INTRODUCTION TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS ... · 7. Time dependent equations - Some examples 40 7.1. Intro 40 7.2. Porous medium equation - solutions with compact support

INTRODUCTION TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

SS 2019

Jan-Frederik Pietschmann

Fakultat fur Mathematik, Technische Universitat ChemnitzReichenhainer Straße 41, Chemnitz, Germany

uncorrected versionPlease send corrections/typos/suggestions to [email protected]

Contents

1. Introduction 22. Some examples of nonlinear PDEs 32.1. The porous medium equation 32.2. Chemical reactions 42.3. Electrons in a semiconductor 42.4. Further examples 52.5. General strategy and reformulation as fixed point problems 53. Fixed point theorems 63.1. Basic setting 63.2. Banach fixed point theory 73.3. Topological fixed point theory 84. Sobolev spaces 114.1. Definition 114.2. Properties 124.3. Convergence and compactness 135. Nonlinear elliptic equations 175.1. Semilinear elliptic equations 175.2. Monotone semilinear equations 235.3. Quasilinear equations 296. Drift-Diffusion equations 356.1. Modeling 356.2. Simplified model & analysis 367. Time dependent equations - Some examples 407.1. Intro 407.2. Porous medium equation - solutions with compact support 407.3. Chemotaxis - finite time blow-up 417.4. Reaction diffusion equations 41References 42

Key words and phrases. Nonlinear partial differential equations, fixed point theorems, semilinear elliptic equations,quasilinear elliptic equations, drift-diffusion equation.

1

2 JAN-FREDERIK PIETSCHMANN

1. Introduction

These notes are heavily using material from [Jue18], [Smi14] (section 2) and I am grateful to theauthors of these lecture notes for their effort.

Let us start by a short overview on linear partial differential equations (PDEs). Let Ω ⊂ Rd, d ≥ 1,open and bounded and fix T > 0. We start by considering

Linear elliptic PDEs: Prototypical for this class of equations is the Laplace/Poisson equation. Weare looking for u : Ω→ R, s.t.

−∆u = f in Ω,

u = g on ∂Ω,

for given data g : ∂Ω→ R (Dirichlet boundary data), or

∂nu = ∇u · n = f,

for given f : ∂Ω→ R (Neumann boundary data), or combinations of both (Robin boundary data.For the general form of elliptic equations, the operator −∆ is replaced by

L := −d∑

i,j=1

aij∂iju+d∑i=1

bi(x)∂iu+ cu,

for given (possibly time dependent) coefficients aij , bj , c : Ω× [0, T ]→ R. We introduce the matrixA := (aij)i,j=1,...,d can say that L is uniformly elliptic if there exists a constant θ > 0 s.t., for all

ξ ∈ RdξTAξ ≥ θ|ξ|2

holds.

Linear parabolic PDEs: These are modelled after the heat equation. We are looking for u : Ω ×[0, T ]→ R, s.t.

∂tu−∆u = f in Ω× (0, T ),

u = g or ∂nu = h on ∂Ω× (0, T ),

for f : Ω× [0, T ]→ R and g, h : ∂Ω× [0, T ]→ R and with initial condition

u(x, 0) = u0(x), u0 : Ω→ R.

Again, we can generalize by replacing −∆ by the elliptic operator L, i.e.

∂tu+ Lu = f in Ω× (0, T ),

where the equation is called parabolic whenever L is elliptic.

Linear hyperbolic PDEs: In this case, we have two prototypical equations. For the transport equa-tion, we are looking for u : Ω× [0, T ]→ R, s.t.

∂tu+ v · ∇u = 0 in Ω× (0, T ),

with a given velocity field v : Ω × [0, T ] → Rd; Note that in this case one has to be more carefulwith boundary conditions. They can only be presecibed on the inflow part

∂Ω− := x ∈ ∂Ω : v(x, t) · n(x) < 0,

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with n beeing the outward normal. The second equation in the wave equation given as

∂ttu = c∆u in Ω× (0, T ),

where c : Ω× [0, T ]→ R is the sound speed that gives how fast the waves can travel.

In this lecture, we will (mostly) focus on nonlinear elliptic and parabolic equations

2. Some examples of nonlinear PDEs

2.1. The porous medium equation. We want to obtain an equation that describes the evolutionof a fluid in a porous medium, e.g. water intruding a dam or water in a rock layer. We are thuslooking for an equation for the fluid density u(x, t). Our first assumption is that u satisfies thecontinuity equation

∂tu+∇ · (uv) = 0,

where v denotes the average fluid velocity. In what follows, we will always assume all quantities tobe scaled. The above equation is called conservation law since the total mass

∫R3 u dx is constant

in time, i.e. conserved. Next we assume that the velocity v is proportional to the gradient of apreasure p. This is the so-called Darcy’s law

v = −k∇p,

and k > 0 depends on the viscocity of the fluid and the permeability of the mediumThe third assumption is that the preasure is given by the following state equation

p = nγ , γ > 0.

Combining this, we obtain the porous medium equation

∂tu−∇ · (kγ

γ + 1∇nγ+1) = 0,

which we can rewrite in the more general form

∂tu−∇ · (D(n)∇n) = 0

with D(n) = kγnγ . We are dealing with a so-called quasilinear parabolic equation. In general, wewant to solve this equation on an open and bounded Ω ⊂ Rd and wee need to supplement it withinitial- and boundary-conditions, e.g.

u = 0 auf ∂Ω, t > 0,

u(·, 0) = u0(·) in Ω

The function D(u) can be seen as a density dependent diffusion coefficient. The larger the fluiddensity, the larger the diffusivity becomes. However, the differential operator

Lu = −∇ · (D(u)∇u)

is not uniformly elliptic (as D(0) = 0) and we are thus dealing with a degenerate parabolic equation.Note that if D(u) is constant, i.e. D(u) = D0, we recover the heat equation which is known tohave a infinite speed of propagation. This is in contrast to the porous medium equations which hasfinite propagation speed.


2.2. Chemical reactions. We want to model the concentration of a chemical substance u = u(x, t)in a domain Ω. We start with

∂tu−D0∆u = R+ +R−, in Ω× (0, T ),

where we assumed constant diffusivity D0 > 0. The function R+ describes sources (of u) while R−models sinks. We assume that the chemical is produced with a constant rate R+ = R0 > 0 andfurther choose R− = −u2 ≤ 0. This models a binary reaction, e.g.

A+A→ B,

in which two molecules of chemical A are reacting to form one molecule of chemical B. We add thefollowing initial and boundary conditions

∂u

∂n= 0 auf ∂Ω, t > 0, u(·, 0) = u0 in Ω.

This constitutes a semilinear parabolic equation of the form

∂tu−∆u = f(x, u).

Such equations are also called reaction-diffusion equations.What happens as t → ∞? We expect that for large times the concentration u becomes constant,i.e. ∂tu = 0. This yields

−∆u = R0 − u2 in Ω,∂u

∂n= 0 auf ∂Ω,

which can be solved explicitly as u∞ =√R0.

In chemical applications, it is of particular interest to know how fast u(, t) converges towards u∞,e.g. if one can expect exponential convergence

‖u(·, t)− u∞‖ ≤ ‖u0‖ e−λt

and if yes, for which rate λ > 0 (and in which norm).

2.3. Electrons in a semiconductor. The motion of charged particles (say electrons) in a semi-conductor is classically (as opposed to quantum mechanically) described by a flux density wherechanges in the particle density n are due to diffusion and drift by the electric field. This yields thetotal flux density

J = ∇n− n∇φ,where φ denotes the electrical potential and −∇φ the electrical field. The evolution of the electrondensity is then given by

∂tn = ∇ · J = ∇ · (∇n− n∇φ).

In the following, we are only interested in stationary solutions and thus assume ∂tn = 0, i.e.

∇ · J = ∇ · (∇n− n∇φ) = 0.

The electrical potential φ is the solution to the Poisson equation

∆φ = n− f,

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where f : Ω → R denotes given, fixed charges in the system and thus n − f is the total chargedensity. The paar (n, φ) is thus solution to the system

∇ · (∇n− n∇φ) = 0,

∆φ = n− f(x)

within the domain of the semiconductor Ω. This is a system of nonlinear elliptic PDEs, the so-calledstationary drift-diffusion model. As boundary conditions we prescribe, on ∂Ω, the electron densityand the electrical potential (i.e. applied voltage):

n = g, φ = ψ auf ∂Ω

In applications one is interested in the mapping ψ 7→∫K J · n ds, where K ⊂ ∂Ω is the electrode

through which the voltage is applied. This is the so-called current-voltage characteristic of thesemiconductor.

2.4. Further examples. While all examples above a of second order, there are also many examplesof nonlinear PDEs with different order. Very important are first order equations, in particularhyperbolic conservation laws of the form

∂tu+∇ · F (u) = 0,

where one prominent example is Burger’s equation which, in 1d, is given as

∂tu+ ∂x(1

2u2) = 0,

or Hamilton-Jacobi equations of the form

∂tu+ F (∇u) = 0.

An example for an equation of third order is the Korteweg-de-Vries equation

∂tu+ u∂xu+ ∂xxxu = 0

which can be analysed using special techniques that are not part of this lectureOne main difference of these examples (compared to second order equations) is the lack of diffusiveeffects which yields less regular solutions. On the other hand, diffusion is not limited to secondorder equations as is illustrated by the fourth-order equation with logarithmic nonlinearity, theso-called Derrida-Lebowitz-Speer-Spohn equation

∂tu+ ∂xx(u∂xx(ln(u))) = 0.

Here, u describes the density of electrons in a specific semiconductor. End first lecture (4/4/19)

2.5. General strategy and reformulation as fixed point problems. Let us outline one ofthe one of the most common strategies to tackle nonlinear problems on the example of

∂tu−∆u = f(x, u).

It consists of the following steps:

(1) Assuming that we know how to solve linear equations we make the above problem linearby replacing u on the right hand side by a given u which yields

∂tu−∆u = f(x, u).

(2) Solving this linear problem yields, for given u yields a solution u to the above equation. Inother words, it defines an operator T : u 7→ u.


(3) Clearly, a fixed point of T would be solution to the original nonlinear problem. But in orderto apply classic fixed-point theorems (e.g. Banach, Schauder, Brouwer, see below), we needestimates on u, independent of u. These so-called a priori estimates are often the mostdifficult part.

This strategy motivates the more detailed study of fixed-point theorems in the next section.

3. Fixed point theorems

In the following, we only consider real spaces

3.1. Basic setting.

Definition 3.1.1 (topological space). A topological space is a pair (X, τ) consisting of a set X anda subset τ ∈ 2X s.t.

(1) ∅, X ∈ τ(2) If U1, . . . , Un ∈ τ , n ∈ N, then

⋂ni=1 Ui ∈ τ .

(3) For Λ an arbitrary set and Uλ ∈ τ for all λ ∈ Λ, we have⋃λ∈Λ

Uλ ∈ τ.

Now we define what we mean by a fixed point

Definition 3.1.2. Let X be a topological space and let T : X → X be a map. A point x ∈ X is afixed point if T (x) = x.

Let us also recall the following definitions:

Definition 3.1.3 (metric space). A metric space is a pair (X, d) where X is a set and d a mappingd : X ×X → [0,∞), called metric, with the following properties:

• d(x, y) = 0 iff (if and only if) x = y.• d(x, y) = d(y, x)• d(x, y) ≤ d(x, z) + d(z, y), x, y, z ∈ X (triangle inequality)

Definition 3.1.4 (normed space). A normed space (X, ‖.‖) is a pair that consists of a vectorspaceX and a mapping ‖ · ‖ : X → [0,∞) s.t.

• ‖x‖ = 0 iff x = 0.• ‖λx‖ = |λ|‖x‖ for all λ ∈ R, x ∈ X..• ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ X.

Remark 3.1.5. Every metric space induces a topological space with open sets defined as

Bε(x) := y ∈ X : d(x, y) < ε,

and every normed space induces a metric one by means of

d(x, y) = ‖x− y‖.

In the sequel, we will investigate under which conditions on X and T the existence of a (possiblyunique) fixed point can be assured. Recall also the notion of a Banach space

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Definition 3.1.6. A normed vector space X is a Banach space if the metric space (X, d) is com-plete, where

d(x, y) := ‖x− y‖ for all x, y ∈ X,denotes the metric induced by the norm.

Most common example: Rd, with the norm given by the Euclidean distance. Another example:Space of continuous, real-valued functions C(X;R) with norm

‖f‖C(X;R) := supx∈X|f(x)| for all f ∈ C(X;R).

3.2. Banach fixed point theory.

Definition 3.2.1. Let (X, d) be a metric space and T : M ⊂ X → X be a map. We say T is acontraction if, for all x, y ∈M with x 6= y, there exists k ∈ (0, 1) such that

d(T (x), T (y)) ≤ kd(x, y).

Theorem 3.2.2 (Banach’s fixed point theorem). Let (X, d) be a complete metric space and M ⊂ Xnonempty and closed. If a map T : M →M is a contraction, then T has a unique fixed point x ∈M .Furthermore, the error estimate

d(x, xn) ≤ kn

1− kd(x1, x0)

holds.

Proof. Note that closed subsets of complete metric spaces are also complete metric spaces, so it issufficient to consider the case M = X. Fix some point x0 ∈ X and define a sequence (xn)x∈N byxn+1 = T (xn). Then

d(x2, x1) = d(T (x1), T (x0)) ≤ kd(x1, x0),

for some k ∈ (0, 1). Continuing inductively gives

d(xn+1, xn) ≤ knd(x1, x0).

Thus, for n < m, we have

d(xn, xm) ≤ d(xn, xn+1) + · · ·+ d(xm−1, xm)

≤ (kn + kn+1 + · · ·+ km−1)d(x1, x0)

= kn(1 + k + · · ·+ km−n−1)d(x1, x0)

≤ kn

1− kd(x1, x0),

(1)

where we have made use of the triangle inequality and the properties of geometric sums (using|k| < 1). Again as |k| < 1, kn/(1 − k) → 0 as n → ∞. Hence (xn)n∈N is Cauchy and has a limitx ∈ X by completeness. Contraction maps are continuous, so it follows that

T (x) = limn→∞

T (xn) = limn→∞

xn+1 = x,

as desired.To see that the fixed point x ∈ X is unique, suppose there is x′ 6= x in X such that x′ is also a fixedpoint. Then d(T (x), T (x′)) = d(x, x′), since both T (x) = x and T (x′) = x′. But d(T (x), T (x′)) <d(x, x′) since T is a contraction, a contradiction.Passing to the limit m→∞ in (1) yields the desired error estimate.


Remark 3.2.3. The requirement d(T (x), T (y)) ≤ kd(x, y) with k < 1 is essential; merely d(T (x), T (y)) <d(x, y) would not be enough.

Banach’s fixed point therorem can be used to prove many important results e.g. the inverseand implicit mapping theorems or the Picard-Lindelof existence theorem for ordinary differentialequations. In the context of this lecture, it will be used in the context of nonlinear parabolicequations.

3.3. Topological fixed point theory.

Brouwer fixed point theory.

Notation. In Rd denote by | · | the Euclidean norm, the closed unit ball in by Bd := B1(0) = x ∈Rd : |x| ≤ 1 and the unit sphere (the boundary of the unit ball) by Sd−1 := x ∈ Rd : |x| = 1 =∂Bd.

Definition 3.3.1. Let A be a subset of a topological space X. A retraction is a map r : X → Asuch that r(x) = x for all x ∈ A. If there exists a retraction from X to A, we say A is a retract ofX.

Theorem 3.3.2 (No-Retraction Theorem). There is no continuous retraction r : Bd → Sd−1.

For d = 1, the boundary of the interval [−1, 1] is just the points −1, 1 which are not connected.As [−1, 1] is connected, no continuous map r : [−1, 1]→ −1, 1 can exist.For d = 2, a possible candidate for a retract could be r(x) = r/|x|, yet we see that this functionhas a singularity at x = 0.Intuitively, there is no function that continuously “makes room” for every mapped point from theinterior of the sphere. Proving the no-retraction theorem for n-dimensional space, however, is notas trivial as it might seem. The most common methods make use of tools far out of the scope ofthis lecture, so we will simply assume Lemma 3.3.2. Proofs using algebraic topology can be founde.g. in [Hat02].

Theorem 3.3.3 (Brouwer’s Fixed Point Theorem). Every continuous map T : Bd → Bd has afixed point.

Proof. Suppose there exists a map T : Bd → Bd with no fixed points. Construct the map r : Bd →Sd−1 by extending a ray along the path from x to T (x) and defining r(x) to be the intersection ofthe ray with the sphere Sd−1 (see Figure 1). Explicitly, this map is given as

r(x) := x+

√1− |x|2 +

⟨x,

x− T (x)

|x− T (x)|

⟩2

−⟨x,

x− T (x)

|x− T (x)|

⟩ x− T (x)

|x− T (x)|

The map r is well-defined since x 6= T (x) for any x ∈ Bd, and continuous since T is continuous.Moreover, r(x) = x for all x ∈ Sd−1, so r is a retraction from Bd to Sd−1. But this contradictsLemma 3.3.2, which says that no such retraction exists. Hence T must have a fixed point.

Corollary 3.3.4. Let K be a nonempty, compact and convex subset of Rd. Then every continuousmap T : K → K has a fixed point.

The general idea behind corollary 3.5 is that K is homeomorphic (i.e. there exits a map f : K → Bd

which is a bijection, continuous, and with its inverse function f−1 being continuous) to the closedunit ball Bd. For a detailed proof, see e.g. [Nie10].

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T(x)xr(x)

Figure 1. Sketch of the retraction

Remark. For now, everything was finite dimensional.

End 2nd lecture, 11.4.

Schauder fixed point theory.

Definition 3.3.5. Let X be a normed vector space and F = x1, x2, . . . , xn a finite subset of X.Then conv(F ), the convex hull of F , is defined by

conv(F ) = n∑j=1

tjxj :n∑j=1

tj = 1, tj ≥ 0.

For future applications, we will need a more general definition to handle the case in which F isinfinite:

Definition 3.3.6. Let X be a normed vector space and F a subset of X. The convex hull conv(F )is the intersection of all convex sets S ⊂ X such that F ∈ S.

Lemma 3.3.7. Definitions 3.3.5 and 3.3.6 are equivalent for finite sets.

Definition 3.3.8 (ε-net). Let ε > 0. A subset S of a metric space X is called ε-net of X if

X ⊂⋃x∈S

Bε(x).

A metric space X is said to be totally bounded if there is a finite ε-net for all ε > 0.

Remark. Instead of totally bounded, the term relatively compact is also sometimes used. Thismay, however, also mean that the closure of the set is compact. Only for complete metric spaces,both notions coincide.

In order to derive the next fixed point theorem, we need the following tool:

Lemma 3.3.9 (Schauder Projection Lemma). Let K be a compact subset of a normed vector spaceX, with metric d induced by the norm ‖ · ‖. Given ε > 0, there exists a finite subset F ⊂ X anda map P : K → conv(F ) such that d(P (x), x) < ε for all x ∈ K. This map is called the Schauderprojection.


Proof. Take a finite ε-net for the compact setK and denote by F = x1, . . . , xn the set of midpointsof the ε-balls. For i = 1, . . . , n, define functions φi : K → R by

φi(x) :=

ε− d(x, xi) x ∈ Bε(xi),0 otherwise.

We see that φi is strictly positive on Bε(xi) and vanishes elsewhere. Therefore∑n

i=1 φi(x) > 0 forall x ∈ K. We define the Schauder projection P : K → conv(F ) by

P (x) =

n∑i=1

φi(x)

φ(x)xi, where φ(x) =

n∑i=1

φi(x).

The map P is continuous since all the φi are. Moreover,

d(P (x), x) =

∥∥∥∥∥n∑i=1

φi(x)

φ(x)xi −

n∑i=1

φi(x)

φ(x)x

∥∥∥∥∥ =

∥∥∥∥∥n∑i=1

φi(x)

φ(x)(xi − x)

∥∥∥∥∥≤

n∑i=1

φi(x)

φ(x)‖xi − x‖ <

n∑i=1

φi(x)

φ(x)ε = ε,

because φi(x) = 0 if ‖xi − xk‖ ≥ ε.

Theorem 3.3.10 (Schauders Fixed Point Theorem). Let X be a Banach space and let M ⊂ X benonempty, convex, and closed. If T : M → M is compact (i.e. T (M) is relatively compact), thenT has a fixed point.

Proof. Let K = T (M) denote the closure of T (M) which, by hypothesis, is compact. For eachn ∈ N, let Fn be a finite 1

n -net for K (which exists as K is compact) and let

Pn : K → conv(Fn)

be the corresponding Schauder projection. The convexity of M implies that conv(Fn) ⊂M so thatwe can define the finite dimensional mapping

Tn : conv(Fn)→ conv(Fn), Tn := (Pn T )|conv(Fn) .

Corollary 3.3.4 guarantees that Tn has fixed points. For each n ∈ N , we choose one such fixedpoint of Tn and call it xn. Since K is compact (xn)n∈N has a convergent subsequence, which wedenote (xnk)k. This sequence converges to some x ∈ K as k → ∞, which we claim is the desiredfixed point. From Lemma 3.3.9 we obtain

d(T (x), xnk) = d(T (x), Tnk(xnk)) ≤ d(T (x), T (xnk)) + d(T (xnk), Tnk(xnk))

≤ d(T (x), T (xnk)) +1

nk→ 0 as k →∞,

since T is continuous. Thus (xnk)k converges to both x and T (x). Limits are unique, so T (x) = x,as desired.

Clearly, this implies the infinite dimensional version of Corollary 3.3.4, i.e.

Corollary 3.3.11. Let X be a Banach space and let M ⊂ X be nonempty, convex, and compact.If T : M →M is continuous, then T has a fixed point.

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In practice, it is sometimes difficult to apply Schauder’s fixed point theorem as one needs to findan appropriate set M . This motivates an alternative formulation, known as Schaefer’s fixed pointtheorem.

Theorem 3.3.12 (Schaefer’s Fixed Point Theorem). Let X be a Banach space and T : X → X bea continuous and compact mapping. If the set

x ∈ X : x = λT (x) for some λ ∈ [0, 1]is bounded, then T has a fixed point.

Proof. By hypothesis, we can choose a constant M so large that ‖x‖ < M if x = λT (x) for someλ ∈ [0, 1]. Define a retraction r : X → BM (0) by

r(x) :=

x ‖x‖ ≤M,(M/‖x‖)x ‖x‖ > M,

and observe that the composition (r T ) : BM (0) → BM (0) is compact since T is compact. LetK denote the closed convex hull of (r T )(BM (0)). The set K is convex by definition, and thecompactness of r T implies K is compact. By Schauders fixed point theorem, there exists a fixedpoint x ∈ K of the restriction (r T )|K : K → K. We claim that x is also a fixed point of T . Toshow this, it is sufficient to prove that T (x) ∈ K. Suppose not. Then ‖T (x)‖ > M and

x = r(T (x)) =M

‖T (x)‖T (x),(2)

which implies

‖x‖ =

∣∣∣∣ M

‖T (x)‖T (x)

∣∣∣∣ = M.

On the other hand, by assumption, ‖x‖ < M , a contradiction.

End third lecture (18/4/19)

4. Sobolev spaces

Most results in this section are provided without proof. See, e.g. [Eva98] or [AF03].

4.1. Definition.

Definition 4.1.1 (Weak derivative). Let

u, v ∈ L1loc(Ω) := w ∈ L1(Ω′) for all Ω′ ⊂ Ω, Ω′ compact,

and let α be a multiindex (i.e. α = (α1, . . . , αd) and |α| = α1 + . . .+ αd. We say that v is the α-thweak partial derivative of u, written as

Dαu = v,

provided ∫ΩuDαϕ dx = (−1)|α|

∫Ωvϕ dx.

holds for all ϕ ∈ C∞c (Ω).

Remark. One can show that the weak derivative is unique up to a set of measure zero, cf. [Eva98,Section 5.2].


Definition 4.1.2. Take m ∈ N0 and 1 ≤ p ≤ ∞. The Sobolev space W k,p(Ω) is the set of allfunctions u ∈ Lp(Ω), s.t.

Dαu ∈ Lp(Ω) for all |α| ≤ k,where α is a multi-index and Dαu the respective weak derivative. We identify functions that coincidea.e. (!) and endow these spaces with the norms

‖u‖pWk,p(Ω)

=∑|α|≤k

‖Dαu‖pLp(Ω) , for p <∞ and

‖u‖Wk,∞(Ω) = max|α|≤k

‖Dαu‖L∞(Ω) .

They are Banach spaces with respect to them.

Remark 4.1.3. In reality, elements of Sobolev spaces are equivalence classes of functions thatcoincide almost everywhere (a.e.), i.e. up to sets of measure zero. In particular, point evaluation(or evaluation on d− 1-dimensional sets) does not make sense a priori.

Remark 4.1.4. For p = 2, we denote Hk(Ω) := W k,2(Ω) which is then a Hilbert space with scalarproduct

(u, v)Hk(Ω) =∑|α|≤k

(Dαu,Dαv)L2(Ω) (e.g. (u, v)H1(Ω) = (u, v)L2(Ω) + (∇u,∇v)L2(Ω))

Definition 4.1.5. Let k ∈ N and 1 ≤ p ≤ ∞. Then we denote by W k,p0 (Ω) the closure of C∞0 (Ω)

with respect to the Norm ‖ · ‖Wk,p(Ω).

Remark. Roughly speaking, these are all functions with

Dαu = 0 on ∂Ω for all |α| < k.

We will make this more precise with the definition of traces below.

4.2. Properties.The following theorem tells us that Sobolev functions can be approximated by smoother functions.

Theorem 4.2.1 (Density). Let 1 ≤ p <∞ and m ∈ N0. We have

C∞(Ω) ∩Wm,p(Ω)⊂

densely Wm,p(Ω)

and

C∞0 (Ω)⊂

densely Wm,p0 (Ω).

If Ω ⊂ R is a bounded domain with ∂Ω ∈ C0,1, then we also have

C∞(Ω)⊂

densely Wm,p(Ω).

Sobolev spaces have the important embedding properties.

Theorem 4.2.2 (Sobolev embedding). Let Ω ⊂ Rd be an open, bounded domain with ∂Ω ∈ C0,1(Ω)and let 1 ≤ p, q <∞, k,m ∈ N0 with m > k.

(1) The embedding Wm,p(Ω) → W k,q(Ω) is continuous, whenever m − n/p ≥ k − n/q andcompact whenever m− n/p > k − n/q.

(2) The embedding Wm,p → Ck(Ω) is compact whenever m− n/p > k.

NONLINEAR PDES 13

For arbitrary domains (i.e. without ∂Ω ∈ C0,1(Ω)), the assertions still holds for W k,p0 (Ω) instead

of W k,p(Ω).

See [Alt16, Sections 10.9, 10.13].

Remark 4.2.3. These results can be extended into various directions:

(1) Less regular domains: E.g. replace Lipschitz by cone condition(2) Refine (2) by embedding into Holder spaces. Indeed, it holds:

• The embedding Wm,p → Ck,α(Ω) is compact whenever m− n/p > k + α and• The embedding Wm,p → Ck,α(Ω) is continuous whenever m − n/p = k + α and α ∈

(0, 1).

In order to be able to give a compact notation in the special case H1(Ω) → Lp(Ω) let us introduce

N∗ =

[1, 2d

d−2

]: d ≥ 3

[1,∞) : d = 2[1,∞] : d = 1

, N∗ =

[

2dd+2 ,∞

): d ≥ 3

(1,∞) : d = 2[1,∞) : d = 1

We call (p, q) an admissible pair if p ∈ N∗ and 1/p+ 1/q = 1. In this case, we automatically haveq ∈ N∗.For example, we have that H1(Ω) → Lp(Ω) is continuous for all p ∈ N∗, i.e. for d = 1, 2,H1(Ω) → Lp(Ω), for every 1 ≤ p <∞ (for d = 1 even in L∞(Ω), by Thm 4.2.2, part (2)), for d = 3H1(Ω) → L6(Ω), and for d = 4 H1(Ω) → L4(Ω), . . . .Also note that the embedding H1(Ω) → L2(Ω) is always compact.

Theorem 4.2.4 (Trace theorem). Let Ω ⊂ Rd be a bounded domain with ∂Ω ∈ C0,1 and let 1 ≤p <∞. Then there exists a continuous, linear map (called trace; dt. Spur), γ : W 1,p(Ω)→ Lp(∂Ω)s.t.

γ(u) = u|∂Ω for all u ∈W 1,p(Ω) ∩ C0(Ω).

Theorem 4.2.5 (Characterization of W 1,p0 functions). Let Ω ⊂ Rd be a bounded domain with

∂Ω ∈ C0,1 and let 1 ≤ p <∞ and u ∈W 1,p(Ω). Then

u ∈W 1,p0 (Ω) iff γ(u) = 0.

Finally, recall the Poincare inequality

Theorem 4.2.6. Let Ω ⊂ Rd be a bounded domain with ∂Ω ∈ C0,1 and let 1 ≤ p <∞. Then thereexists a constant CP > 0, depending only on d, p and Ω s.t.

‖u‖Lp(Ω) ≤ CP ‖∇u‖Lp(Ω) ∀u ∈W 1,p0 (Ω),(P0)

and, for 1 ≤ p ≤ ∞ and u ∈W 1,p(Ω) there exists CP , depending only on d, p and Ω, s.t.

‖u− 1

|Ω|

∫Ωu dx‖Lp(Ω) ≤ CP ‖∇u‖Lp(Ω) ∀u ∈W 1,p

0 (Ω),(P0)

4.3. Convergence and compactness.In finite dimensional spaces, every bounded sequence possess a convergent subsequence (Bolzano-Weierstrass theorem). This is not longer true in infinite dimensional spaces (consider, e.g. an infiniteorthogonal system). This lack of compactness is a common problem when analysing nonlinear PDEsand to overcome this issue, we will introduce the notion of weak convergence. We start by giving(without proof) some functional analytic results (see, e.g. [Bre11]).


Let B always be a (real) Banach space with Norm ‖ · ‖. Its dual space is defined as

B∗ := f : B → R : f is linear and bdd.With the norm

‖f‖B∗ = sup‖u‖≤1

|f(u)| = sup‖u‖≤1

|〈f, u〉B∗ |,

B∗ becomes a Banach space itself. For example, by Riesz-Frechet, the dual of Lp can be identifiedwith Lq whenever 1 ≤ p <∞ and 1/p+ 1/q = 1.Now we introduce the notion of weak convergence.

Definition 4.3.1 (Weak convergence). A sequence (uk)k∈N ⊂ B is said to converge weakly tou ∈ B if for every F ∈ B∗

〈F, uk〉B∗ → 〈F, u〉B∗ as k →∞.In this case we write

uk u as k →∞.

End fourth lecture (25/4/19)

Remark. In Lp-spaces, this specialises to: (uk)k∈N ⊂ Lp(Ω) converges weakly to u ∈ Lp if forevery φ ∈ Lq(Ω) and 1/p+ 1/q = 1 we have∫

Ωukφ dx→

∫Ωuφ dx as k →∞.

Remark 4.3.2. Warning: Based on the definition of weak convergence, one can define the weaktopology using the neighbourhood system

UF,ε = x ∈ X | F (x) < ε for all F ∈ C where C ⊂ X∗.

But if X is not separable, there exists no countable basis and thus compact (dt. Uberdeckungskompakt)and sequencially compact (dt. Folgenkompakt) do not coincide.

Example 4.3.3. We claim that uk(x) := sin(kx)/π converges weakly, in L2((0, 2π)), to zero. Letf ∈ (L2(0, 2π))∗ = L2(0, 2π). Then

(f, uk)L2 =1

π

∫ 2π

0sin(kx)f(x) dx

are the Fourier coefficients of f which, according to the theorem of Riemann-Lebesgue, converge tozero as k →∞. This implies

uk 0 in L2((0, 2π)).

However, uk does not converge strongly in L2((0, 2π)) as

‖uk − 0‖2L2(0,2π) =1

π2

∫ 2π

0sin2(kx) dx =

1

π6= 0

This shows that weak convergence does in general not (!) imply strong convergence (also, ukconverges neither pointwise nor almost everywhere).The concept of weak convergence is useful since we have the following compactness property forBanach spaces (which is a special case of the Theorem of Banach-Alaoglu).

Theorem 4.3.4 (Eberlein-Smulian). Every bounded sequence in a reflexive Banach space has aweakly converging subsequence.

NONLINEAR PDES 15

Remark 4.3.5. The space Lp(Ω) is reflexive iff 1 < p < ∞. It is separable (i.e. contains an atmost countable, dense subset) whenever 1 ≤ p < ∞. So L1(Ω) is separable but not reflexive andhas dual space L∞(Ω). The space L∞(Ω) is neither reflexive nor separable. Its dualspace (L∞(Ω))∗

can be identified with the set of bounded, finitely additive signed measures which is a strict subsetof L1(Ω).

The proof for the following properties of weakly converging sequence can be found, e.g. in [Alt16,Chapter 8]

Proposition 4.3.6 (Properties of weak convergence). Let B be a Banach space. Then

i) Strong convergence implies weak convergence.ii) If dim(B) <∞, then weak convergence also implies strong convergence.

iii) (Thm of Banach-Steinhaus): If uk converges weakly to u (for (uk)k ⊂ B) as k → ∞, then ukis bounded and

‖u‖ ≤ lim infk→∞

‖uk‖ .

iv) If B is reflexive, and all weakly convergent subsequences of a bounded sequence (uk)k ⊂ B havethe same limit u ∈ B, then the complete sequence (uk)k converges weakly to u. (The sameholds true for strong convergence).

Part iv) can be strengthened: IfB is a topological space, then uk converges to u iff every subsequenceof (uk) has another subsequence that converges to the same limit u ∈ B.

Remark 4.3.7. What happens if we compose a weakly converging sequence with a continuousfunction, i.e. if uk u in B and f : B → B is continuous, what can be say about the sequence(f(uk))k? Unfortunately, in general, f(uk) does not converge, neither weak or strong.Take for example uk(x) = sin(kx) for x ∈ (0, 2π), und f(x) = |x|. From example 4.3.3 we knowthat uk 0 in L2(0, 2π). On the other hand, |uk| 0 in L2(0, 2π) cannot hold as, for φ(x) = 1∫ 2π

0| sin(kx)| · 1 dx =

1

k

∫ 2πk

0| sin y|dy =

∫ 2π

0| sin y|dy = 4.

In nonreflexive (yet separable) Banach spaces (e.g. L1(Ω)), there exists an analogue to the Theorem

of Eberlein-Smulian (Theorem 4.3.4), yet based on so-called weak* convergence. This is useful whenwe know that a sequence is bounded in L∞(Ω) (a situation in which Thm 4.3.4 does not apply).

Definition 4.3.8 (weak-∗ convergence). A sequence (Fk)k in B∗ is said to converge weak* toF ∈ B∗ if, for all u ∈ B

〈Fk, u〉B∗ → 〈F, u〉B∗ as k →∞.We write

Fk ∗ F as k →∞.

Remark. In a Hilbert space, weak and weak* convergence coincide.

Theorem 4.3.9. Let B be a separable Banach space. Then every bounded sequence in B∗ posessesa weak* converging subsequence.

For the proof see, e.g. [Zei90, Section 21.8]Let (uk) be a bounded sequence in L∞(Ω) and B = L1(Ω). As (L1(Ω))∗ = L∞(Ω), there existsukj

∗ u in L∞(Ω), u.e. ∫Ωuklv dx→

∫Ωuv dx for all v ∈ L1(Ω)


Proposition 4.3.10 (Properties of weak* convergence). Let B be a separable Banach space.

i) Strong convergence implies weak* convergence.ii) In reflexive spaces weak and weak* convergence are equivalent.

iii) (Thm of Banach-Steinhaus) If (uk)k ⊂ B∗ converges weak* to u ∈ B∗ as k → ∞, then uk isbounded and

‖u‖B∗ ≤ lim infk→∞

‖uk‖B∗ .

iv) If uk ∗ u in B∗ and vk → v in B (or if uk → u in B∗ and vk v in B), then

〈uk, vk〉B∗ → 〈u, v〉B∗ as k →∞.

Remark. Note that as in a Hilbert space, weak and weak* convergence coincide, iv) also holds forthe product of a weak- and a strongly converging sequence

For a bounded sequence (uk) in L1(Ω), neither Theorem 4.3.4 nor Theorem 4.3.9 applies. In general,it is indeed wrong that bounded sequences in L1(Ω) have weakly convergent subsequences - thisonly holds under additional assumptions, e.g. Dunford-Pettis theorem.

Definition 4.3.11. A subset K of L1(Ω) is called uniformly integrable if for every ε > 0 thereexists a δ > 0 s.t. ∫

E|f | dλn(x) < ε, for f ∈ K and λn(E) < δ.

Theorem 4.3.12 (Dunford-Pettis). A subset of L1(Ω) is relatively weakly compact (i.e everybounded sequence has a convergent subsequence, iff it is bounded and uniformly integrable.

Remark 4.3.13. The de la Vallee-Poussin theorem yields the following equivalent characterisationof uniform integrability:

A familiy (un)n∈N ⊂ L1(Ω) is uniformly integrable iff there exists a non-negativeincreasing and convex function φ : [0,∞] → [0,∞) s.t. limx→∞ φ(x)/x = ∞and supk

∫Ω φ(|uk|) dx <∞.

If we chose φ(x) = |x|α with α > 1, we see that the boundedness of ‖uk‖Lα(Ω) yields the existence

of a subsequence (ukn)n with ukn u in L1(Ω).

Finally, recall the theorem about dominated convergence (or Lebesque’s theorem) which yields theopposite assertion:

Theorem 4.3.14 (dominated convergence). Let Ω ⊂ Rd (n ≥ 1) be an open set and (fk)k ⊂ L1(Ω)a sequence.

• If fk → f a.e. in Ω as k → ∞ and there exists a function g ∈ L1(Ω) with |fk| ≤ g a.e. inΩ, then f ∈ L1(Ω) and fk → f in L1(Ω).• If fk → f in L1(Ω), then there exists a subsequence (fkj )j with fkj → f a.e. in Ω and a

function g ∈ L1(Ω) s.t. |fkj | ≤ g for all j.

The theorem remains valid if we replace L1(Ω) by Lp(Ω) where we assume 1 ≤ p <∞ for part (i)and 1 ≤ p ≤ ∞ for part (ii) in [Bre11, Thm IV.9].

End 5th lecture (2/5/19)

Finally, consider the following definition

NONLINEAR PDES 17

Definition 4.3.15. For X and Y two Banach spaces, we say that X is embedded into Y if thereexists a bounded linear operator T : X → Y with Ker(T ) = 0. In particular we have

‖Tu‖Y ≤ C‖u‖X .Furthermore, we call the embedding dense whenever Im(T ) is dense in Y .

Proposition 4.3.16. Let B and X be Banach spaces with continuous and dense embedding B → X.Then the embedding X∗ → B∗ is continuous as well. If X is reflexive, then X∗ is even dense inB∗.

Remark. Recall that B is called reflexive if (B∗)∗ = B.

We will apply the proposition to X = Lp(Ω) and B = H10 (Ω), for p ∈ N∗, and obtain

Lq(Ω) ⊂ H−1(Ω) := (H10 (Ω))∗, q ∈ N∗, 1/p+ 1/q = 1.

5. Nonlinear elliptic equations

5.1. Semilinear elliptic equations. For Ω ⊂ Rd, open, bounded, we consider the model problem

Lu = f(x, u) in Ω, u = g on ∂Ω.(3)

We shall make the following standing assumptions:

(A1) ∂Ω ∈ C1

(A2) The differential operator L is defined as

Lu = −∇ · (A∇u) + cu,

with A(x) = (aij)i,j=1,...d for aij , c : Ω→ R given.(A3) L is uniformly elliptic with constant α > 0, i.e.

ξTAξ ≥ α|ξ|2 for all ξ ∈ Rd a.e. x ∈ Ω.

(A4) aij , c ∈ L∞(Ω) and c ≥ 0.(A5) g is such that it can be extended to H1(Ω) (i.e. such that there exists g ∈ H1(Ω) with

γ(g) = g).

Finally, we ask f to be a Caratheodory function according to the following definition

Definition 5.1.1. We call f : Ω× R→ R a Caratheodory function if

• x 7→ f(x, u) is measurable for all u ∈ R and• u 7→ f(x, u) is continuous for almost all x ∈ R.

Reminder

ab ≤ ε

2a2 +

b2

2ε(a, b > 0, ε > 0) (Cauchy’s inequality with ε)∫

Ωuv dx ≤ ‖u‖Lp(Ω)‖v‖Lq(Ω)

(1 ≤ p, q ≤ ∞, 1/p+ 1/q = 1

u ∈ Lp(Ω), v ∈ Lq(Ω)

)(Holder’s inequality)

‖u‖Lp(Ω) ≤ CP ‖∇u‖Lp(Ω) = CP ‖u‖W 1,p(Ω) (u ∈W 1,p0 (Ω)) (Poincare inequality)

We have the following result on Caratheodory functions


Lemma 5.1.2. Let Ω ⊂ Rd (d ≥ 1) and f a Caratheodory function with

|f(x, u)| ≤ C|u|r + h(x), 1 ≤ q, qr <∞, C ≥ 0, h ∈ Lq(Ω), h ≥ 0,

for a.e. x ∈ Ω and u ∈ R. Define F (u)(·) = f(·, u(·)) for u ∈ Lqr(Ω). Then, F is a continuousmapping from Lqr(Ω) to Lq(Ω).

Proof. Take (uk)k ⊂ Lqr(Ω) with uk → u in Lqr(Ω). Now take an arbitrary subsequence ukj whichalso converges to u. Using the reverse of the dominated convergence theorem (Thm 4.3.14) thereexists a further subsequence (ukjl )l s.t. ukjl → u a.e. in Ω as l → ∞, and such that the uniform

estimates |ukjl | ≤ u∗ ∈ Lqr(Ω) holds for all j. In particular, this implies F (ukjl )→ F (u) a.e. in Ω.

As ∣∣∣F (ukjl)∣∣∣q ≤ C (|u∗|qr + hq) ∈ L1(Ω),

we know that |F (ukjl )|q is dominated by an L1-function and we can apply the dominated conver-

gence theorem to show that F (ukjl )→ F (u) in Lq(Ω), as l →∞. The assertion then follows from

the proceeding lemma, applied to (F (uk))k∈N.

Lemma 5.1.3. Let (un)n be a sequence in a topological space X, such that every subsequence (unk)khas a further subsequence (unkm )m converging to u ∈ X. Then the full sequence converges to x.

Proof. Assume (un)n does not converge to u. That means that u has a neighbourhood U such thatA = n : un /∈ U is infinite. Enumerating A in ascending order produces a subsequence (unk)that has no term is in U . By the assumptions, (unk) has a subsequence (unkm )m converging to u.That means xnkm ∈ U for all large enough m, contradicting the construction unk /∈ U for all k.Hence the assumption was wrong, and un → u.

First we state the following assumption

(S1) f is a Caratheodory function with

|f(x, u)| ≤ h(x) for x ∈ Ω, u ∈ R, where h ∈ Lq(Ω), q ∈ N∗,and introduce the weak formulation of (3): Find u ∈ H1(Ω) with γ(u) = g on ∂Ω s.t.∫

Ω

(∇uTA∇v + cuv

)dx =

∫Ωf(x, u)v dx holds for all v ∈ H1

0 (Ω)(4)

Note that the integral on the right hand side is well-defined due to the embedding H10 (Ω) → Lp(Ω)

(for p ∈ N∗) which yields v ∈ Lp(Ω) and, using Holder’s inequality with 1/p+ 1/q = 1 gives

f(·, u)v ∈ L1(Ω)

Under these assumptions, we have the following existence result:

Theorem 5.1.4 (Existence for semilinear equations). Under the assumptions (A1)-(A5) and (S1)there exists a weak solutions u ∈ H1(Ω) (in the sense of (4)) to (3) and a constant C > 0, dependingonly on A, c and Ω, such that

‖u‖H1(Ω) ≤ C(‖g‖H1(Ω) + ‖h‖Lq(Ω)

).(5)

Proof. We divide the proof into several steps:

Step 1: Definition of the fixed point operator. Let v ∈ K where

K :=v ∈ L2(Ω) : ‖v‖H1(Ω) ≤M

,

NONLINEAR PDES 19

and where M > 0 will be fixed later on. To show that K is compact in L2(Ω) since for everysequence (uk)k ⊂ K, using the compact embedding H1(Ω) → L2(Ω) (cf. Theorem 4.2.2), there

exists a subsequence (ukj ) that s.t. ukj → u in L2(Ω). Also (Thm. of Eberlein-Smulian) ukj u

in H1(Ω) as j →∞. The limit u is in K as, due to Proposition 4.3.6, we have

‖u‖H1(Ω) ≤ lim infkl→∞

‖ukl‖H1(Ω) ≤M,

and thus, K is compact.


Now let u ∈ H1(Ω) be the unique weak solution to the linear boundary value problem

Lu = f(x, v) in Ω, u = g on ∂Ω.(6)

We know that such a weak solution exists (Due to the Theorem of Lax-Milgram, cf. e.g. [Eva98,Section 6.2]), whenever the right hand side is in H−1(Ω) (the dual space of H1

0 (Ω)). This is truesince due to Proposition 4.3.16 we have

|f(·, v(·))| ≤ h(·) ∈ Lq(Ω) → H−1(Ω).

Thus we can define the operator

S : K → L2(Ω), v 7→ u.

It remains to show that S is a selfmapping (i.e. S : K → K) and that it is continuous.Step 2: S is self-mapping. We show that S(K) ⊂ K. To this end consider u = S(v) ∈ S(K).

Chosing the test function u− g ∈ H10 (Ω) in the weak form of (6) yields∫

Ω

(∇uTA∇(u− g) + cu(u− g)

)dx =

∫Ωf(x, v)(u− g) dx(7)

We start by estimating the left hand side using the ellipticity of A and Cauchy’s inequality with ε(ab ≤ (ε/2)a2 + b2/(2ε)) as follows∫

Ω

(∇uTA∇(u− g) + cu(u− g)

)dx

=

∫Ω

(∇(u− g)TA∇(u− g) +∇gTA∇(u− g) + c(u− g)2 + cg(u− g)

)dx

≥∫

Ω

(α|∇(u− g)|2 − α

2|∇(u− g)|2 − |A|

2

2α|∇g|2 + c(u− g)2 − c

2(u− g)2 − c

2g2

)dx

≥ α

2

∥∥∇(u− g)2∥∥2

L2(Ω)− 1

2α‖A‖2L∞(Ω)‖∇g‖

2L2(Ω) −

1

2‖c‖L∞(Ω)‖g‖2L2(Ω)

(8)

In order to estimate the right hand side of (7), we use the assumptions on f and Holder’s inequality:∫Ωf(x, v)(u− g) dx ≤

∫Ωh|u− g| dx ≤ ‖h‖Lq(Ω)‖u− g‖Lp(Ω) ≤

1

2ε‖h‖2Lq(Ω) +

ε

2‖u− g‖2Lp(Ω),

with 1/p + 1/q = 1. Then, as p ∈ N∗ and we have H1(Ω) → Lp(Ω). Using the continuity of theembedding and Poincare’s inequality implies

‖u− g‖Lp(Ω) ≤ C1‖u− g‖H1(Ω) = C1

(‖u− g‖L2(Ω) + ‖∇(u− g)‖L2(Ω)

)≤ C1(CP + 1)‖∇(u− g)‖L2(Ω).


Now if we choice ε > 0 sufficiently small, e.g. ε = α/(2C21 (CP + 1)2), we have

ε

2‖u− g‖2Lp(Ω) ≤

α

4‖∇(u− g)‖2L2(Ω),

and thus can absorb this term (i.e. subtracting it on both sides of the weak formulation) using theequivalent term (yet with prefactor α/2) in (8). Summarising, we have

α

4‖∇(u− g)‖2L2(Ω) ≤

1

2α‖A‖2L∞(Ω)‖∇g‖

2L2(Ω) +

1

2‖c‖L∞(Ω)‖g‖2L2(Ω) +

C22

α‖h‖2Lq(Ω) =: M2

0

Using Poincare’s inequality one more time then yields

‖u‖H1(Ω) ≤ ‖u− g‖H1(Ω) + ‖g‖H1(Ω) ≤(C3M0 + ‖g‖H1(Ω)

)=: M,(9)

and we conclude u ∈ K. This shows that the operator is self-mapping and implies the a prioriestimate (5), as soon as we know the existence of a fixed point.Step 3: S is continuous. Take a sequence (vk)k ⊂ K s.t. vk → v in L2(Ω) as k → ∞. We have to

show that S(vk)→ S(v) in L2(Ω). To this end chose a subsequence (vkj )j and denote uj = S(vkj ),i.e. uj satisfies ∫

Ω

(∇uTj A∇w + cujw

)dx =

∫Ωf (x, vj)w dx for w ∈ H1

0 (Ω)

and in particular, by (9), we know that (uj)j is uniformly bounded in H1(Ω). Thus there exists asubsequence (ujl)l = (S(vkjl ))l, so that for l→∞

ujl u inH1(Ω), ujl → u in L2(Ω),

where the first convergence is a consequence of Eberlein-Smulian (Thm. 4.3.4), the second of thecompact embedding of H1 into L2. We obtain∫

Ω

(∇uTjlA∇w + cujlw

)dx→

∫Ω

(∇uTA∇w + cuw

)dx as l→∞

Using the continuity of the trace operator γ : H1(Ω)→ L2(∂Ω) and γ(ujl) = g for all l we see thatujl u in H1(Ω) implies γ(ujl) γ(u) in L2(∂Ω), i.e. γ(u) = g (The property that for a weaklyconverging subsequence, the image is also weakly converging is called weak sequential continuityand only holds for linear, continuous mappings between Banach spaces).It remains to carry out the limit in the term

∫Ω f(x, vkj )w dx. According to Lemma 5.1.2 the

mapping v 7→ f(·, v) is continuous from L2(Ω) to Lq(Ω) (chose C = 0 and r = 2/q). Thus vkj → v

in L2(Ω) implies f(·, vkj )→ f(·, v) in Lq(Ω) and thus∫Ωf(x, vkj

)w dx→

∫Ωf(x, v)w dx.

In particular, this also holds for the subsequence vkjl . Therefore, the limit u satisfies∫Ω

(∇uTA∇w + cuw

)dx =

∫Ωf(x, v)w dx, w ∈ H1

0 (Ω).

and is thus a weak solution to the problem Lu = f(x, v) in Ω with u = g on ∂Ω. As this problemis uniquely solvable, we conclude u = S(v) and thus we have shown that

S(vkjl ) = uj → u = S(v) in L2(Ω) as l→∞.

NONLINEAR PDES 21

The limit u is unique and thus by Lemma 5.1.3, the whole sequence S(vk) converges to u and thecontinuity of S is proven.Step 4: Existence of a fixed point. We see that all assumptions needed for Schauder’s fixed point

theorem (Theorem 3.3.10) are satisfied and thus there exists u ∈ H1(Ω) s.t. u = S(u). Clearly,this u is a (weak) solution to (3).


Since the structure of this proof will repear frequently from now on, let us again summarise:

Proof via Schauder’s fixed point argument

(1) Define a set K = u ∈ L2(Ω) : ‖u‖H1(Ω) ≤M ⊂ L2(Ω) and show that K is compact.(2) Pick v ∈ K and condider the linear problem Lu = f(x, v).(3) This defines an operator S : K → L2(Ω) with Sv = u.(4) Show that, for all v ∈ K, we have uniform bound ‖u‖H1(Ω) ≤M , i.e. S(K) ⊂ K.(5) Short that S : K → K is continuous.(6) Apply Schauder’s fixed point theorem to S

Remark 5.1.5. We implicitly showed that for u ∈ H10 (Ω), the norms ‖∇u‖L2(Ω) and ‖u‖H1(Ω) =

‖∇u‖L2(Ω) + ‖u‖L2(Ω) are equivalent by Poincare’s inequality. Indeed

‖∇u‖L2(Ω) ≤ ‖u‖H1(Ω) ≤ (1 + CP )‖∇u‖L2(Ω),

1

2min(1, 1/CP )‖u‖H1(Ω) ≤ ‖∇u‖L2(Ω) ≤ ‖u‖H1(Ω).

Remark 5.1.6. The calculation in the proof above become simpler if we assume homogenous bound-ary conditions, i.e. g = 0. The general case can be reduced to this one by means of the transfor-mation v = u− g (g ∈ H1(Ω) s.t. γ(g) = g), with u being a solution to

Lu = f(x, u) in Ω u = g on ∂Ω.

Then, v solves

Lv = Lu− Lg = f(x, v + g(x))− Lg =: F (x, v) in Ω, v = 0 on ∂Ω.

The problem is that F does no longer satisfy the assumptions of theorem 5.1.4 as Lg ∈ H−1(Ω).However, the proof remains almost unchanged if we only ask for the following, weaker assumptions

(S2) Let F (·, v) = F1(·, v) + F0 with F0 ∈ H−1(Ω) and F1 Caratheodory with |F1(·, v)| ≤ h ∈Lq(Ω), q ∈ N∗.

Thus we will from now on mostly assume g = 0 to simplify the proofs. Based on the followingtheorem, we shall see that solutions to (3) can be more regular that H1, namely two times weaklydifferentiable if the coefficients and data are sufficiently smooth (cf. [GT01, Thm. 9.13, 9.14]). Weneed the following regularity result on linear elliptic PDEs.

Theorem 5.1.7 (W 2,r regularity of linear elliptic equations). Let assumptions (A1) to (A5) bevalid and assume furthermore ∂Ω ∈ C2, aij ∈ C0,1(Ω), g ∈W 2,r(Ω) and f ∈ Lr(Ω) for 1 < r <∞.Then there exists exactly one solution u ∈W 2,r(Ω) to

Lu = f in Ω, u = g on ∂Ω,


together with the a-priori estimate

‖u‖W 2,r(Ω) ≤ C(‖g‖W 2,r(Ω) + ‖f‖Lr(Ω)

)for some constant C > 0, independent of u.

Corollary 5.1.8. Let the assumptions of Theorems 5.1.4 and 5.1.7 hold. Then there exists a weaksolution u ∈W 2,q(Ω) to (3).

Proof. We know the existence of a weak solution u to Lu = f(x, u) in Ω with u− g ∈ H10 (Ω). Given

this solution, we interpret the right hand side of (3) as a function

f(x) := f(x, u(x)).

In particular, u is then a weak solution to Lu = f and as |f | ≤ h ∈ Lq(Ω), Theorem 5.1.7 isapplicable and implies u ∈W 2,q(Ω).

We may now ask the question into which directions we can generalise Theorem 5.1.4, e.g.

• Whether the solution u is indeed unique, or• A solution still exists for more general nonlinearities of the form |f(x, u)| ≤ C|u|β + h(x),

for some β > 0.

Unfortunately, in general, both questions have a negative answer; the following examples showsthis for uniqueness

Example 5.1.9 (Nonuniqueness). Consider the boundary value problem

−u′′ = cu in (0, 1), u(0) = u(1) = 0,

with c > 0 (note that c ≤ 0 is included in Theorem 5.1.4). First we observe that u = 0 is a solutionfor all c > 0. From the theory of ordinary differential equations we know, on the other hand, thatthere exist nontrivial solutions u 6= 0 whenever c = (2πk)2 for k ∈ N. Thus, for these choices of cwe have nonuniqueness.

There are, however, also situations in which the above questions have a positive answer. Consider,e.g. the nonlinearity f to be Lipschitz continuous, i.e.

|f(x, u)− f(x, v)| ≤ f0|u− v| for a.e. x ∈ Ω, all u, v ∈ R,

with Lipschitz constant f0 > 0 being “sufficiently small”. In this case, uniqueness can be shown(based on a fixed point argument).As for more general nonlinearities, they can be admitted as long as they are sublinear, i.e. if for0 < β < 1, C > 0 and h ∈ Lq(Ω) there holds

|f(x, u)| ≤ C|u|β + h(x) for x ∈ Ω, u ∈ R(10)

The critical step in the proof of Theorem 5.1.4 is to show the uniform estimates on u in H1(Ω).Assume g = 0 and consider solutions u ∈ H1

0 (Ω) to Lu = f(x, v) for given v ∈ H10 (Ω). Choosing u

itself as test function in the weak formulation and using (10) yields∫Ω

(α|∇u|2 + cu2

)dx ≤

∫Ωf(x, v)u dx ≤ C

∫Ω|v|β|u| dx+

∫Ωh|u| dx.

NONLINEAR PDES 23

Applying Holder’s and Young’s inequality to both terms on the right hand side we estimate∫Ω

(α|∇u|2 + cu2

)dx ≤ C‖v‖β

Lβq(Ω)‖u‖Lp(Ω) + ‖h‖Lq(Ω)‖u‖Lp(Ω)

≤ ε‖u‖2Lp(Ω) +C2

2ε‖v‖2β

Lβq(Ω)+

1

2ε‖h‖2Lq(Ω),

where 1/p+ 1/q = 1 and q ∈ N∗. Then p ∈ N∗ and H1(Ω) → Lp(Ω). For ε sufficiently small, e.g.ε = α/(2CP ) we apply Poincare’s inequality and conclude

‖u‖H1(Ω) ≤ C1

(‖v‖β

Lβq(Ω)+ ‖h‖Lq(Ω)

)≤ C2

(Mβ + ‖h‖Lq(Ω)

),

since ‖v‖Lβq(Ω) ≤ C3‖v‖H1(Ω) ≤ C3M as v ∈ K. To ensure H1(Ω) → Lβq(Ω), we chose (e.g. for

d ≥ 3) p = 2d/(d − 2) and q = 2d/(d + 2). Then H1(Ω) → Lβq(Ω), whenever −d/(βq) ≤ 1 − d/2or β ≤ 2d/((d− 2)q) = (d+ 2)/(d− 2), which is satisfied as β < 1. Now we need to find M > 0 s.t.

C2

(Mβ + ‖h‖Lq(Ω)

)≤M,

as this ensures u ∈ K and this is possible iff β < 1. We have shown that S : K → K.

From an application point of view, semilinear equations can be used to describe stationary state ofchemical reactions (cf. Section 2.2). In this case, we often have superlinear growth of the nonlinearright hand side. It is still possible to prove existence in this case, yet not with the methods presentedabove. Thus we only state a result and refer to [Eva98, Section 8.5.2] for details and the proof.

Theorem 5.1.10. Let Ω ⊂ Rd be open and bounded with ∂Ω ∈ C1 and 1 < p < (d + 2)/(d − 2)(p <∞ if d ≤ 2). Then there exists a weak solution u ∈ H1

0 (Ω), u 6= 0 to

−∆u = |u|p−1u in Ω, u = 0 on ∂Ω

Remark 5.1.11. Thus the previous theorem implies nonuniqueness, as u = 0 is always a secondadmissible solution. One can show that if p > (d + 2)/(d − 2), then u = 0 is the only (classical)solution, at least if Ω is star shaped around 0, cf. [Eva98, Section 9.4.2]. We call u = 0 the trivialsolution and p = (d+ 2)/(d− 2) the critical exponent.


5.2. Monotone semilinear equations. We expect that for monotone decreasing (and maybesuperlinear) functions f(x, u), one can still show boundedness of u and thus existence of solutions.As in the previous section, we consider equations of the type

Lu = f(x, u) in Ω, u = g on ∂Ω,(11)

yet now with the additional assumption

(S3) Let f : Ω× R→ R be s.t. the function u 7→ f(x, u) is monotone decreasing (for a.e. x).

We have the following result.

Theorem 5.2.1 (Existence and Uniqueness for monotone, semilinear equations). Let Assumption(A1) - (A5) hold, let f be a Caratheodory function s.t. (S3) holds. Furthermore, let

|f(x, u)| ≤ c|u|p−1 + h(x) for a.e. x ∈ Ω, u ∈ R,(12)


hold where c > 0, h ∈ Lq(Ω), q ∈ N∗, 1 < p < 2d/(d − 2) (p < ∞, if d ≤ 2) and 1/p + 1/q = 1.Then there exists exactly one weak solution to (11) and a constant C > 0 that only depends on A,c and Ω s.t.

‖u‖H1(Ω) ≤ C(‖g‖H1(Ω) + ‖h‖Lq(Ω)

)Note that the values of p and q are chosen s.t. the embedding H1(Ω) → Lp(Ω) is compact andthat f(·, u) ∈ Lq(Ω) holds. This is a consequence of

|f(·, u)|q ≤ C(|u|(p−1)q + hq

)= C (|u|p + hq) ∈ L1(Ω)

The proof will again be based on a fixed point argument, yet this time we cannot use Schauder’sfixed point theorem, as it will destroy the monotonicity of the r.h.s.. Indeed, assuming g = 0 andfor given v, let u be a solution to the linearised equation Lu = f(x, v). As in the proof of Theorem5.1.4, this defines a fixed point operator S(v) = u. Taking now, in the weak formulation of thelinearised equation, the function u as a testfunction, we obtain

α

∫Ω|∇u|2 dx ≤

∫Ω|f(x, v)u| dx(13)

We have to estimate the r.h.s.. If we would deal with the term f(x, u)u instead of f(x, v)u, themonotonicity of f would imply

f(x, u)u− f(x, 0)u = (f(x, u)− f(x, 0))(u− 0) ≤ 0

and thus we could estimate

f(x, u)u ≤ f(x, 0)u ≤ ε

2u2 +

1

2εf(x, 0)2

The term f(x, 0)2 is independent of u while the remaining term ε2u

2 could be absorbed by the lefthand side of (13) (after an application of Poincare’s inequality). Unfortunately, these argumentsare not valid for f(x, v)u, i.e. the fixed point argument distroys the monotonicity. To circumventthis issue we use the following alternative fixed point theorem.

Theorem 5.2.2 (Leray-Schauder). Let B be a Banach space, S : B × [0, 1] → B a compact andcontinuous mapping with S(v, 0) = 0 for all v ∈ B. Furthermore, there exists a C > 0 s.t. for allu ∈ B and σ ∈ [0, 1] with S(u, σ) = u there holds

‖u‖ ≤ C.

Then, v 7→ S(v, 1) has at least one fixed point.

Remark. Note that Schaefer’s fixed point Theorem (Thm (3.3.12)) is in fact a special case ofLeray-Schauder and that Leray-Schauder is also a consequence of Schauder’s fixed point theorem(Thm. 3.3.10), cf. [GT01, Thm 11.6] for the proof. As for Schaefer, the advantage is that we donot have to explicitly construct a compact and convex set. Instead, the fixed point operator has tobe compact and all fixed points have to satisfy a uniform bound.

Another advantage is that the estimates only have to hold for fixed points. Applied to (11) (withg = 0), we can use u as a test function and obtain, as sketched above,

α

∫Ω|∇u|2 dx ≤

∫Ωf(x, u)u dx ≤

∫Ωf(x, 0)u dx ≤ ε

2‖u‖2L2(Ω) +

1

2ε‖f(·, 0)‖2L2(Ω).

NONLINEAR PDES 25

which yieldsα

2‖∇u‖2L2(Ω) ≤ C(α)‖f(·, 0)‖2L2(Ω),

i.e. a uniform bound for u.

Remark. This is often called a priori estimate, since at this point, it is not yet clear that such anu exists.

Proof of Thm 5.2.1. (The structure is very similar to the proof of Thm 5.1.4).Step 1: Definition of the fixed point operator. For given v ∈ Lp(Ω) and σ ∈ [0, 1] let u ∈ H1(Ω) →Lp(Ω) be the unique weak solution to

Lu = σf(x, v) in Ω, u = σg on ∂Ω(14)

This defines the operator

S : Lp(Ω)× [0, 1]→ Lp(Ω), u = S(v, σ).

For σ = 0, the solution to (14) is u = 0, i.e. S(v, 0) = 0 for all v ∈ Lp(Ω)Step 2: S is continuous and compact (Similar to Step 3, proof of Thm. 5.1.4.) Let vk with ‖vk‖Lp(Ω) ≤K be a bounded sequence s.t. vk → v in Lp(Ω) and σk → σ as k →∞. Define uk = S(vk, σk) anduse uk − σkg as a testfunction in (the weak form of) (14):∫

Ω

(∇uTkA∇ (uk − σkg) + cuk (uk − σkg)

)dx = σk

∫Ωf (x, vk) (uk − σkg) dx.

As in the proof of Theorem 5.1.4, we can estimate the left hand side using the ellipticity of A, sothat

α

2

∫Ω|∇uk − σkg|2 dx ≤

∫Ω|f (x, vk) (uk − σkg)| dx+ C(A, c, g)

≤ ε

2‖uk − σkg‖2Lp(Ω) +

1

2ε‖f (·, vk)‖2Lq(Ω) + C(A, c, g)

(15)

where 1/p+1/q = 1. The constant C(A, c, g) depends on the L∞-norms of A and c and on the H1-norm of g and we have used Holder’s- and Young’s inequality in the last step. Using the continuityof the embedding H1 → Lp(Ω), Poincare inequatility and σk ≤ 1, we can estimate the first termon the r.h.s. as

ε

2‖uk − σkg‖2Lp(Ω) ≤

ε

2C2

1 ‖uk − σkg‖2H1(Ω) ≤ 2

ε

2C2

1 (1 + CP )2 ‖∇uk − σkg‖2L2(Ω)

For ε sufficiently small, e.g. ε = α/(4C21 (1 + CP )2), we can absorb the first term on r.h.s by the

left side of (15). The second term is estimated as follows, using (12):∫Ω|f (x, vk)|q dx ≤ c

(∫Ω|vk|(p−1)q dx+

∫Ω|h|q dx

)≤ C3(K2 + ‖h‖Lq(Ω))

since (p− 1)q = p. Thus right hand side is bounded which implies

‖uk‖H1(Ω) ≤ C4.(16)

Thus, the sequence uk = S(vk, σk) is bounded in H1(Ω) and compact in Lp(Ω), i.e there exists asubsequence s.t.

ukj u in H1(Ω) and ukj → u in Lp(Ω) as j →∞.


Since vk → v in Lp(Ω), we also have that f(x, vkj ) → f(x, v) in Lq(Ω)(see Lemma 5.1.2). Thustaking the limit j →∞ in the weak formulation∫

Ω

(∇ukj

tA∇w + cukjw)dx = σk

∫Ωf(x, vkj )w dx, w ∈ H1

0 (Ω)

then yields ∫Ω

(∇utA∇w

)dx = σ

∫Ωf(x, v)w dx, w ∈ H1

0 (Ω)

where, due to weak continuity of the trace operator, the limit also satisfies the boundary conditions.Thus, u is a solution to (14), i.e. u = S(v, σ) and S(vkj , σkj ) = ukj → u = S(v, σ) in Lp(Ω). Asour arguments are still valid if we start with an arbitrary subsequence and due to the uniquenessof the limit, Lemma 5.1.3 again shows the convergence of the whole sequence S(vk, σk). Thus, S iscontinuous.The compactness of S is a consequence of the (uniform in k) estimate (16). Indeed, if (vk)k∈Nis a bounded sequence in Lp(Ω), then (uk)k∈N = (S(vk, σk))k∈N is bounded in H1(Ω), i.e. has aconverging subsequence in Lp(Ω).Step 3: A priori estimate. It remains to show the estimates on the fixed points. To this end, let

σ ∈ [0, 1] and u ∈ H1(Ω) be a fixed point of S(·, σ). We take u− σg ∈ H10 (Ω) as a testfunction and

obtain (by a similar calculation as above)

α

2

∫Ω|∇u|2 dx ≤ σ

∫Ωf(x, u)(u− σg) dx+ C(A, c, g)

= σ

∫Ω

(f(x, u)− f(x, σg))(u− σg) dx

+ σ

∫Ωf(x, σg)(u− σg) dx+ C(A, c, g)

(17)

Using the monotonicity of f(x, ·) and σ ≤ 1 then implies

α

2

∫Ω|∇u|2 dx ≤ ε

2‖u− σg‖2Lp(Ω) +

1

2ε‖f(·, σg)‖2Lq(Ω) + C(A, c, g)

≤ ε

2‖u‖2Lp(Ω) +

1

2ε‖f(·, σg)‖2Lq(Ω) + C(A, c, g)

(18)

Again, we apply Sobolev-, Triangle-, and Poincare inequatility to the first term on the r.h.s andobtain, for ε sufficiently small

ε

2‖u‖2Lp(Ω) ≤

α

4‖∇u‖2L2(Ω) + C‖g‖2H1(Ω)

The first term on the right hand side can be absorbed by the left hand side of (17). The secondterm in (18) is estimated using (12), so that we obtain

α

4

∫Ω|∇u|2 dx ≤ C5

(‖g‖2(p−1)

L(p−1)q(Ω)+ ‖h‖2Lq(Ω)

)+ C(A, c, g)

which is bounded as (p − 1)q = p and due to the continuous embedding H1(Ω) → Lp(Ω). Weconclude the (uniform in the choice of fixed point) estimate

‖u‖Lp(Ω) ≤ C6‖u‖H1(Ω) ≤ C7,

where C7 > 0 depends on g and h. We are now in a position to apply the fixed point theoremof Leray-Schauder (Thm 5.2.2) to conclude the existence of a fixed point, i.e. a solution to (11).

NONLINEAR PDES 27

Step 4: Uniqueness. Let u1 and u2 be two weak solutions to (11). Then, u1 − u2 ∈ H10 (Ω) is a

admissible test function and we obtain∫Ω

(∇uTi A∇ (u1 − u2) + cui (u1 − u2)

)dx =

∫Ωf (x, ui) (u1 − u2) dx, i = 1, 2

subtracting the two equations and using the monotony of f however yields∫Ω

(∇ (u1 − u2)T A∇ (u1 − u2) + c (u1 − u2)2

)dx =

∫Ω

(f (x, u1)− f (x, u2)) (u1 − u2) dx ≤ 0,

and, using the ellipticity of A, we conclude ∇(u1 − u2) = 0 in Ω, i.e. u1 − u2 = const a.e. in Ω. Asu1 − u2 has zero Dirichlet boundary conditions, we conclude u1 = u2.

We can avoid growth conditions on f if we look for solutions that are bounded in L∞(Ω). Thisis possible, assuming f to be monotone, if we assume additionally that g ∈ L∞(Ω). To show this(essential) boundedness of the solutions, we use a variant of the maximum principle. To this end,we first extend the classical maximum principle to a maximum principle for weak solutions. Thiswill be based of

Theorem 5.2.3 (Stampacchia). Let Ω ⊂ Rd (d ≥ 1) an open, bounded set, 1 ≤ p < ∞ andu ∈W 1,p(Ω). Then, u+ := max(u, 0) ∈W 1,p(Ω) and

∇u+ =

∇u if u > 0,0 otherwise.

Furthermore, we have that ∇u = 0 a.e. in u = 0 := x ∈ Ω : u(x) = 0.

The proof of this theorem is based on the following Lemma:

Lemma 5.2.4. Let F ∈ C1(R) with F ′ ∈ L∞(R) and u ∈ W 1,p(Ω), 1 ≤ p < ∞. Then, F u ∈W 1,p(Ω) and

∇(F u) = F ′(u)∇u.

Proof of Lemma 5.2.3. As F (x) = x+ is not a C1-function, we have to regularise it (cf., e.g. [Tro13,Thm 1.56]). Consider

Fε(x) :=

√z2 + ε2 − ε x > 0,

0 otherwise.

Then Fε ∈ C1(R) and Fε(x)→ x+ as ε→ 0 and for all x ∈ R. Furthermore, Lemma 5.2.4 impliesFε u ∈W 1,p(Ω). Thus, for v ∈ C∞0 (Ω) partial integration yields∫

Ω(Fε u)∇v dx = −

∫ΩF ′ε(u)∇uv dx = −

∫u>0

u∇u√u2 + ε2

v dx.

Thus the limit ε→ 0 on both sides we obtain∫Ωu+∇v dx = −

∫u>0

∇uv dx,

i.e. u+ ∈W 1,p(Ω) and ∇u+ = χu>0∇u. To show the second statement, note that u = u+ − (−u)+

and thus

∇u = χu>0∇u+ χ−u>0∇u = 0 a.e. in u = 0.


f(x, u)

u−M0

M0

Figure 2. Sketch of the situation of Assumption (S4)

Theorem 5.2.5 (Weak maximum principle). Let assumptions (A1)–(A5) hold and let u ∈ H1(Ω)satisfy (in the weak sense)

Lu ≤ 0 in Ω, u ≤ 0 on ∂Ω.

Then, u ≤ 0 a.e. in Ω

Proof. Since u ≤ 0 on ∂Ω, we have u+ = 0 on ∂Ω. The Theorem of Stampacchia ensures that u+

is in H10 (Ω) and thus can be used as a testfunction in the weak formulation. We obtain

0 ≥∫

Ω

(∇uTA∇u+ + cuu+

)dx =

∫u>0

(∇uTA∇u+ cu2

)dx ≥ 0.

As c ≥ 0, and using the ellipticity of A, we conclude ∇u+ = 0 a.e. in Ω, i.e. u+ = const a.e. in Ω.In particular, it satisfies the elliptic equation

∆u+ = 0 in Ω, u+ = 0 on ∂Ω,

which is uniquely solvable with solution u+ = 0. This implies u ≤ 0 in Ω.

This allows us to show existence for semilinear systems without a global growth condition on thenonlinearity. We impose the following assumption

(S4) f : Ω× R→ R is a Caratheodory function s.t.– u 7→ f(x, u) is monotone decreasing for a.e. x ∈ Ω,– There exists M0 > 0 s.t. f(x,M0) ≤ 0 and f(x,−M0) ≥ 0 for a.e. x ∈ Ω,– For all |u| ≤ M := max(M0, ‖g‖L∞(Ω)), there holds |f(x, u)| ≤ h(x) ∈ Lq(Ω) for a.e.x ∈ Ω and q ∈ N∗.

See Figure 2 for a sketch.

NONLINEAR PDES 29

Remark. Note that this is only a local condition. Examples which satisfy (S4) are f(x, u) = −u3

or f(x, u) = 1− eu.

Theorem 5.2.6. Under the assumptions (A1)-(A5) and (S4) there exists a weak solution u ∈H1(Ω) ∩ L∞(Ω) to (11) and a constant C > 0, depending only on A, c and Ω, such that

‖u‖L∞(Ω) ≤ maxM0, ‖g‖L∞(Ω), ‖u‖H1(Ω) ≤ C(‖g‖H1(Ω) + ‖h‖Lq(Ω)

).(19)

Proof. We define

fM (x, v) :=

f(x, v) −M ≤ v ≤M,f(x,M) v > M,f(x,−M) v < −M.

Then, v 7→ fM (x, v) is monotone decreasing, fM (x,M) ≤ fM (x,M0) ≤ 0, fM (x,−M) ≥ fM (x,−M0) ≥0 for M ≥M0 and |fM (x, v)| ≤ max|f(x,M)|, f(x,−M) for all v ∈ R, a.e. x ∈ Ω. Thus, Theorem5.2.1 implies the existence of a weak solution u ∈ H1(Ω) to

Lu = fM (x, u) in Ω, u = g on ∂Ω.(20)

We claim that this function u is already a solution to the original problem (11). To this end, wehave to show that |u| ≤M and thus fM (x, v) = f(x, v).By definition of M we have (u −M)+ = 0 on ∂Ω so that we can use it as a test function in theweak formulation of (20). We obtain∫

Ω

(∇uTA∇(u−M)+ + cu(u−M)+

)dx =

∫ΩfM (x, u)(u−M)+ dx

=

∫Ω

(fM (x, u)− fM (x,M)) (u−M)+ dx+

∫ΩfM (x,M)(u−M)+ dx ≤ 0

since fM is monotone decreasing with fM (x,M) ≤ 0. Stampacchia’s Theorem (Thm. 5.2.3) impliesthat

∇(u−M)+ = χu>M∇(u−M) = χu>M∇u.We thus obtain∫

Ω

((∇(u−M)+

)TA∇(u−M)+ + c

((u−M)+

)2)dx ≤ −

∫ΩcM(u−M)+ dx ≤ 0,

and conclude (u −M)+ = 0, i.e. u ≤ M . The inequality u ≥ −M is obtained analogously bychosing (u+M)− = −(−u−M)+ as a test function.


5.3. Quasilinear equations. In this section we are interested in equations of the form

−∇ · a(∇u) = f in Ω, u = 0 on ∂Ω.(21)

We will follow the same strategy as in [Eva98, Section 9.1] and make the following standing as-sumption on the vector field a:

(Q1) a = (a1, . . . , ad) : Rd → Rd is a continuous vector field.

We also introduce the following notion of monotonicity:


Definition 5.3.1. Let a : Rd → Rd be a given vector field. We call a monotone, whenever

(a(p)− a(q)) · (p− q) ≥ 0 for all p, q ∈ Rd.

We call a strongly monotone, if there exists a constant γ > 0 s.t.

(a(p)− a(q)) · (p− q) ≥ γ|p− q|2 for all p, q ∈ Rd.(22)

Example 5.3.2. For a given, convex and two times continuously differentiable function φ : Rd →R we claim that a := ∇φ is monotone. This is the case since Taylor’s theorem (with integralremainder) implies

(a(p)− a(q)) · (p− q) =d∑i=1

(∂iφ(p)− ∂iφ(q)) (pi − qi)

=d∑

i,j=1

∫ 1

0∂2ijφ(p+ θ(q − p))dθ (pi − qi) (pj − qj) ≥ 0,

where ∂i = ∂/∂pi und ∂2ij = ∂2/∂pi∂pj and where we used that the Hessian D2φ of the convex

function φ is positive definite. If we ask for φ to be uniformly convex, i.e. pTD2φp ≥ γ|p|2 for allp ∈ Rd and some γ > 0, then a is even strongly monotone.One such example is φ(p) =

√1 + |p|2 which yields a(p) = p/

√1 + |p|2 and (inserted into (21))

results in the minimal surface equation.

Again, we introduce a notion of weak solution to (21): We are looking for u ∈ H10 (Ω) with a(∇u) ∈

L2(Ω) and such that ∫Ωa(∇u) · ∇v dx =

∫Ωfv dx for all v ∈ H1

0 (Ω).

We also pose the following additional assumption

(Q2) Assume that a growths at most linearly and is coercive, i.e. there exist constants C > 0,α > 0 and β ≥ 0 s.t.

|a(p)| ≤ C(1 + |p|) and a(p) · p ≥ α|p|2 − β for all p ∈ Rd

Theorem 5.3.3 (Existence for quasilinear equations). Let Assumptions (A1)-(A5) hold and let abe monotone and satisfy (Q1), (Q2). Then, there exists a weak solution u ∈ H1

0 (Ω) to (21).

The growth condition in (Q2) ensures that a(∇u) ∈ L2(Ω), whenever u ∈ H1(Ω). The inequalityin (Q2) corresponds to a generalised coercivity since in the case of the Laplace operator, i.e.a(∇u) = ∇u we clearly have a(p) ·p = |p|2 und thus, the inequality is satisfied for α = 1 and β = 0.The proof is again based on a fixed point strategy, yet this time we use a different approach asbefore: We first define approximate solutions by means of a Galerkin projection. These solutionsare contained in a finite dimensional space to that we can use Brouwer’s fixed point theorem (Thm.3.3.3) without the need of any compactness. Only in the next step, we show that these solutionsare uniformly bounded in terms of the dimension of the finite dimensional space and eventuallypass to the limit of infinite dimensions. This last step again requires a compactness argument.We start by proving the following technical result

NONLINEAR PDES 31

Lemma 5.3.4. Let v : Rd → Rd continuous and let

v(x) · x ≥ 0 for all |x| = r,

for some r > 0. Then there exists x0 ∈ Br(0) s.t. v(x0) = 0.

Proof. Assume that for all x ∈ Br(0) v(x) 6= 0 and define the continuous mapping w : Br(0) →∂Br(0) by

w(x) = − r

|v(x)|v(x), x ∈ Br(0).

We can interpret w as a function from Br(0) to Br(0) and according to Brouwer’s fixed point

theorem (Thm 3.3.3) there exists x0 ∈ Br(0) s.t. w(x0) = x0. The definition of w implies x0 ∈∂Br(0) and thus

r2 = |x0|2 = w (x0) · x0 = − r

|v (x0)|v (x0) · x0 ≤ 0.

As x0 6= 0 this is a contradiction.

Proof of Thm. 5.3.3.Step 1: Definition of a finite dimensional problem. Let (wk)k be an orthonormal basis of H1

0 (Ω)w.r.t the scalar product

(u, v)H10 (Ω) =

∫Ω∇u · ∇v dx.

(For example, wk could be the normalised Eigenfunctions of −∆ in H10 (Ω)). We are looking for

solutions um that are linear combinations of w1, . . . , wm and satisfy the Galerkin equations∫Ωa (∇um) · ∇wk dx =

∫Ωfwk dx for k = 1, . . . ,m.(23)

We can also express the unknown um in this basis as

um =m∑k=1

dkwk

and thus have to use the m Galerkin equations to determine the coefficients dk. Due to the terma(∇u), this yields a nonlinear system of equations which we handly by uing Lemma 5.3.4. Definev = (v1, . . . , vm) : Rm → Rm via

vk(d) =

∫Ω

a m∑j=1

dj∇wj

· ∇wk − fwk dx, k = 1, . . . ,m, d = (d1, . . . , dm) ∈ Rm


If d∗ is a critical point of v, then u =∑m

k=1 d∗kuk is a solution to (23). Now the coercivity of a (see

(Q2)) implies

v(d) · d =

∫Ω

a m∑j=1

dj∇wj

· m∑k=1

dk∇wk − fm∑k=1

dkwk

dx

≥∫

Ω

α∣∣∣∣∣∣m∑j=1

dj∇wj

∣∣∣∣∣∣2

− β − fm∑k=1

dkwk

dx

= α|d|2 − βmeas(Ω)−m∑k=1

dk

∫Ωfwk dx,

since the wk are orthonormal w.r.t (·, ·)H10 (Ω). The weighted Young’s inequality yields

v(d) · d ≥ α

2|d|2 − βmeas(Ω)− 1

2α

m∑k=1

(f, wk)2L2 .

It remains to estimate the last term. To this end denote by φ ∈ H10 (Ω) the weak solution to

−∆φ = f in Ω. Then

(φ,wk)H10

=

∫Ω∇φ · ∇wk dx =

∫Ωfwk dx

and thusm∑k=1

(f, wk)2L2 =

m∑k=1

(φ,wk)2H1

0≤ ‖φ‖2H1

0 (Ω) ≤ C1‖f‖2L2(Ω),

since wk has H10 (Ω)-norm one. We obtain

v(d) · d ≥ α

2|d|2 − βmeas(Ω)− C1

2α‖f‖2L2(Ω).

If we chose r > 0 sufficiently large, this implies v(d) · d ≥ 0 for |d| = r and Lemma 5.3.4 yields theexistence of d∗ with v(d∗) = 0 and thus um =

∑mk=1 d

∗kwk is a solution to (23).


Step 2: A priori estimates. The proof is completed if we can show that the limm→∞ um exists andis a weak solution to (21). Thus we need estimates on um that do not depend (are uniform in) m.We multiply (23) by d∗k and sum over k = 1, . . . ,m which yields∫

Ωa (∇um) · ∇um dx =

∫Ωfum dx.(24)

The coercivity of a implies∫Ω

(α |∇um|2 − β

)dx ≤

∫Ωa (∇um) · ∇um dx =

∫Ωfum dx ≤ ε

2

∫Ωu2m dx+

1

2ε

∫Ωf2 dx

We chose ε = α/CP and use Poincare’s inequality to aborb the first term on the r.h.s. by the l.h.s.and obtain

‖um‖H1(Ω) ≤ C(1 + ‖f‖L2(Ω)

),

the desired uniform estimate.

NONLINEAR PDES 33

Step 3: Limit m→∞. According to the theorem of Eberlein-Smulian (Thm. 4.3.4) there exists a

subsequence (uml)l s.t. uml u in H1(Ω) as l→∞. In particular

∇uml ∇u in L2(Ω)d.(25)

In general, however, we know that weak convergence is not enough to pass to the limit in thesequence a(∇uml). However, based on a technique called the Minty Browder Trick, we can showthat

a (∇um′)→ a(∇u) in L2(Ω)d.

First, the growth condition (Q2) for a and (25) yield that a(∇um) is bounded in L2(Ω), i.e.

‖a (∇um)‖L2(Ω) ≤ C(

1 + ‖∇um‖L2(Ω)

)≤ C.

Thus there exists a subsequence (umlj )j of (uml)l s.t.

a(∇umlj ) b in L2(Ω).

We have to show that b = a(∇u). Taking the limit j →∞ in the Galerkin equations (23) yields∫Ωb · ∇wk dx =

∫Ωfwk dx, k ∈ N.

As (wk)k forms a basis of H10 (Ω), this relation holds for all v ∈ H1

0 (Ω):∫Ωb · ∇v dx =

∫Ωfv dx.(26)

Using the monotonicity of a and (24) implies

0 ≤∫

Ω

(a(∇umlj

)− a(∇v)

)·(∇umlj −∇v

)dx

=

∫Ω

(fumlj − a

(∇umlj

)· ∇v − a(∇v) ·

(∇umlj −∇v

))dx

,

where v ∈ H10 (Ω). The trick where is that we replace the term a(∇umlj ) · ∇umlj , whose limit we

cannot determine directly by fumlj , using (24). Then the above convergence results allow us to

pass to the limit k →∞ which yields

0 ≤∫

Ω(fu− b · ∇v − a(∇v) · (∇u−∇v)) dx.(27)

Now choosing v = u in (26) we can replace fu by b · ∇u and obtain

0 ≤∫

Ω(b− a(∇v)) · (∇u−∇v) dx, for all v ∈ H1

0 (Ω).

Fixing w ∈ H10 (Ω) and v = u± λw for λ > 0. Then

0 ≤ ∓∫

Ω(b− a(∇u± λ∇w)) · ∇w dx.

Taking the limit λ→ 0 then yields

0 ≤ ∓∫

Ω(b− a(∇u)) · ∇w dx,


i.e. the equality

0 =

∫Ω

(b− a(∇u)) · ∇w dx for all w ∈ H10 (Ω).(28)

thus we can replace the term ∫Ωb · ∇v dx by

∫Ωa(∇u) · ∇v dx

in (26) which shows that the limit u satisfies the weak formulation of (21) and thus ends theproof.

Finally, we can also show uniqueness of solutions to (21), if a is strongly monotone.

Theorem 5.3.5 (Uniqueness for quasilinear equations). Let the assumption of Theorem 5.3.3 holdand let a be strongly monotone. Then there exists exactly one solution u ∈ H1

0 (Ω) to (21).

Proof. Assume u1 and u2 are two different weak solutions to (21), i.e.∫Ωa (∇ui) · ∇v dx =

∫Ωfv dx, i = 1, 2.

We substract them and chose v = u1 − u2 as testfunction in the difference to obtain

0 =

∫Ω

(a (∇u1)− a (∇u2)) · (∇u1 −∇u2) dx ≥ γ∫

Ω|∇u1 −∇u2|2 dx,

where we used to the strong monotonicity (22). This implies ∇(u1 − u2) = 0 a.e. in Ω and asu1 − u2 = 0 on ∂Ω it implies u1 − u2 = 0 in Ω.

This result suggest that one can also prove a comparison theorem which is indeed true.

Proposition 5.3.6. Let the assumption of Theorem 5.3.3 hold and let a be strongly monotone. Weset Lu = −∇ · a(∇u). If u, v ∈ H1(Ω) are two functions s.t.

Lu ≤ Lv in Ω, u ≤ v on ∂Ω,

then, u ≤ v a.e. in Ω.

Proof. The inequality Lu ≤ Lv means, in weak formulation,∫Ωa(∇u) · ∇w dx ≤

∫Ωa(∇v) · ∇w dx for all w ∈ H1

0 (Ω)withw ≥ 0.

This can be rewritten as ∫Ω

(a(∇u)− a(∇v)) · ∇w dx ≤ 0.

By assumption the function w = (u − v)+ = max0, u − v satisfies w = 0 on ∂Ω. The Theoremof Stampacchia (Thm 5.2.3) implies that w ∈ H1

0 (Ω) so that w is an admissible test function. Thestrong monotonicity thus implies

0 ≥∫u>v

(a(∇u)− a(∇v)) · (∇u−∇v) dx ≥ γ∫u>v

|∇u−∇v|2 = γ

∫Ω

∣∣∇(u− v)+∣∣2 dx.

Again, we conclude ∇(u − v)+ = 0, i.e. (u − v)+ = const a.e. in Ω. As (u − v)+ = 0 on ∂Ω, wehave (u− v)+ = 0 a.e. in Ω and thus also u ≤ v a.e. in Ω.

NONLINEAR PDES 35

Remark 5.3.7. Unfortunately, the uniqueness result (Thm 5.3.5) result cannot be applied to theminimal surface equation

∇ ·

(∇u√

1 + |∇u|2

)= f in Ω, u = 0 auf ∂Ω,

as a(p) = p/√

1 + |p|2 is not coercive. It is, however, possible to show existence and uniqueness ofsolutions, see [Giu84].


6. Drift-Diffusion equations

We consider the stationary drift–diffusion equations that models the motion of charges particles insemiconductors:

Diffusion

Diffusionp-dotiert n-dotiert

Defektelektron Elektron

+–––

++

Raumladungen

Raumladungszone/Sperrschicht

a)

b)

Figure 3. Sketch of a diode (https://de.wikipedia.org/wiki/Diode)

6.1. Modeling. Consider for example a diode made out of silicon and with added impurities toallow for the motion of negative (”electrons”) and positive (”holes”) charge carriers. We denotetheir respective densities by n = n(x, t) and p = p(x, t). The move due to diffusion and drift bythe electrical fields present in the semiconductor, i.e.

∂tn = ∇ · Jn = ∇ · (Dn∇n− µnn∇V )

∂tp = ∇ · Jp = ∇ · (Dp∇p+ µpp∇V )in Ω× (0, T ),(29)

with initial conditions

n(x, 0) = n0(x), p(x, 0) = p0(x) a.e. in Ω.(30)

Here Dn and Dp denote the diffusion coefficients of positively and negatively charged particles whileµn and µp denote their mobility The electric potential is given by permanent charges f = f(x)present in the system but is also generated by the charge carriers themselves. This is modelled bythe Poisson equation of electrostatics

−ε∆V = e(p− n+ f) in Ω,(31)

where ε denotes the permittivity constant of the material and e denotes the elementary charge.Finally, we have to add boundary conditions. Usually, the diode has a part which is isolated

https://de.wikipedia.org/wiki/Diode


and one on which electrical contacts are attached. Thus we assume that ∂Ω = ∂ΩI ∪ ∂ΩC with∂ΩI ∩ ∂ΩC = ∅ and prescribe

Jn · n = (Dn∇n− n∇V ) · n = 0,

Jp · p = (Dp∇p− p∇V ) · n = 0

∇V · n = 0

on ∂ΩI × (0, T ),(32)

and

p = pD, n = nD, V = VD on ∂ΩC × (0, T ),(33)

for given functions pD, nD and VD. System (29)–(33) is often called Drift-Diffusion equations.Often, it is assumed the processes inside the diode (semiconductor) happen much faster than thechange of the boundary conditions. In this case one neglect the time derivatives in (29) and considerthe stationary system with unknowns n(x), p(x), V (x), only.

6.2. Simplified model & analysis. For the analysis we consider the following simplified modelfor only one species of charged particles u = u(x) and where we set all physical constants to one:

∇ · (∇u− u∇V ) = 0, in Ω,(34)

∆V = u− f, in Ω,(35)

for a given f : Ω→ R which models permanent charges in the system. This is supplemented withthe boundary conditions

u = g, V = ψ on ∂Ω.(36)

From the modelling point of view, u is a particle density (and we thus expect u ≥ 0) and V is theelectrical potential (with −∇V being the electrical field).We are dealing with a system on nonlinear elliptic equations since the expression u∇V is nonlinear(as V depends on u). Again, we will pursue a fixed point strategy to show existence of weaksolutions, based on the following assumptions

(DD1) f ∈ L∞(Ω) and there exist constants f∗ > 0, f∗ > 0 s.t. 0 < f∗ ≤ f ≤ f∗ a.e. in Ω.

(DD2) There exists g, ψ ∈ H1(Ω) ∩ L∞(Ω) s.t. g = γ(g) and ψ = γ(ψ) and constants g∗ > 0,g∗ > 0 s.t. 0 < g∗ ≤ g ≤ g∗ a.e. in ∂Ω.

The weak formulation to (34)–(36) is as follows: Find (u, V ) ∈ (H1(Ω) ∩ L∞(Ω))2 s.t. γ(u) = gand γ(V ) = ψ on ∂Ω and that∫

Ω∇u · ∇v dx =

∫Ωu∇V · ∇v dx,(37) ∫

Ω∇V · ∇w dx = −

∫Ω

(u− f(x))w dx,(38)

holds for all v, w ∈ H10 (Ω). Note that we need u ∈ L∞(Ω) for the integral over u∇V · ∇v to be

well-defined. The L∞-bound on V is not necessary at this point but as we will see later on willanyway be a consequence of our existence result.

Theorem 6.2.1 (Existence for drift–diffusion system). Let assumptions (A1), (DD1) and (DD2)hold. Then there exists a solution (u, V ) ∈ (H1(Ω) ∩ L∞(Ω))2 to (34)–(36) s.t.

1

MeK≤ u ≤MeK , −K ≤ V ≤ K a.e. in Ω,

NONLINEAR PDES 37

whereM = e‖ψ‖L∞(Ω) maxg∗,1/g∗,

M0 = max ln (Mf∗) , ln (M/f∗) ,K = max

M0, ‖ψ‖L∞(Ω)

.

Before we start with the proof, we introduce a change of variables that transforms (34) into asymmetric form. Denote by v := e−V u the so-called Slotboom variable. We obtain

∇v = e−V (∇u− u∇V ) and 0 = ∇ · (∇u− u∇V ) = ∇ ·(eV∇v

),

i.e. we are solving the system

∇ ·(eV∇v

)= 0 in Ω,(39)

∆V = eV v − f(x) in Ω,(40)

with boundary conditions

u = g, V = ψ on ∂Ω.(41)

Every solution (v, V ) ∈ (H1(Ω) ∩ L∞(Ω))2 to (39)–(41) defines, via the transformation u = eV v aweak solution to the original problem (34)–(36). The advantage lies in the fact hat the equationfor v features a symmetric differential operator. This allows us to conveniently derive H1 and L∞

estimates.Typically, these estimates will then depend on V , while V again depends on v via the Poissonequation. We will use a cut-off technique to derive estimates on v independent of V .Finally, we state the following result

Lemma 6.2.2. Let Ω ⊂ Rd be open and bounded, 1 ≤ p <∞ and (uk)k a sequence with uk → u inLp(Ω) as k → ∞ and f ∈ C0(R). If either (uk)k is uniformly bounded in L∞(Ω) or f is boundedin R, then

f(uk)→ f(u) in Lp(Ω).

Proof of Thm. 6.2.1.Step 1: Approximiation via cut off. To obtain L∞-estimates, we use the Stampacchia’s cut-offmethod, i.e. we first consider the system

∇ ·(eV∇v

)= 0, ∆V = eV vM − f(x) in Ω,

with

vM =

1/M : v ≤ 1/Mv : 1/M < v < MM : v ≥M

,

with M as defined above. This ensures that vM ∈ L∞(Ω).Step 2: Construction of a fixed point operator. Given v ∈ L2(Ω) we consider

−∆V = F (x, V ) := −eV vM (x) + f(x) in Ω, V = ψ on ∂Ω.(42)

This is still a semilinear equation and we claim that Theorem 5.2.6 is applicable. Indeed

• F is a Caratheodory function since V 7→ eV is continuous and both vM and f are integrable.• The function V 7→ −eV vM (x) + f(x) is monotone decreasing.


• The definition of vM and M0 imply

F (x,M0) ≤ −eM01

M+ f∗ ≤ 0 and F (x,−M0) ≥ −e−M0M + f∗ ≥ 0.

Thus, all assumptions needed in Theorem 5.2.6 are satisfied and we conclude the existence of aunique solution V ∈ H1(Ω) to (42) with the a priori estimates

‖V ‖L∞(Ω) ≤ K = maxM0, ‖ψ‖L∞(Ω)

, ‖V ‖H1(Ω) ≤ C1(f, ψ,M).(43)

Next we consider, for given σ ∈ [0, 1] the equation

∇ ·(eV∇v

)= 0 in Ω, v = σvD on ∂Ω,

which is a linear elliptic equation with diffusion coefficient A(x) := eV (x). As A(x) ≥ e−K > 0 theproblem is uniformly elliptic and we obtain a unique solution v ∈ H1(Ω) s.t.

‖v‖H1(Ω) ≤ C2(K) ‖vD‖H1(Ω)(44)

Thus we can define the fixed point operator

S : L2(Ω)× [0, 1]→ L2(Ω), S(v, σ) = v,

and immediately conclude that S(v, 0) = 0.


Step 3: Continuity and compactness of S. Let vk, σk be given sequences s.t. vk → v in L2(Ω) andσk → σ as k → ∞ and define vk = S(vk, σk). Furthermore, we denote by Vk the solution to (42)with (vk)M on the r.h.s.. The a priori estimates (43), (44) show that (vk)k and (Vk)k are uniformlybounded in H1(Ω) and, additionally, that (Vk)k is bounded in L∞(Ω). Thus there are subsequencess.t.

vkj v in H1(Ω), vkj → v in L2(Ω)Vkj V in H1(Ω), Vkj → V in L2(Ω)

as j →∞.

Using Lemma 6.2.2 we conclude that eVkj → eV and (vkl)M → vM in L2(Ω). Thus the product

converges in L1(Ω), i.e. eVkj(vkj)M→ eV vM in L1(Ω). In the weak formulation∫

Ω∇Vkj · ∇w dx = −

∫Ω

(eVkj(vkj)M− f(x)

)w dx, w ∈ C∞0 (Ω)

we can pass to the limit j →∞ and obtain∫Ω∇V · ∇w dx = −

∫Ω

(eV vM − f(x)

)w dx, w ∈ C∞0 (Ω).

The same limit yields

0 =

∫ΩeVkj∇vkj · ∇w dx→

∫ΩeV∇v · ∇w dx, w ∈ C∞0 (Ω), as j →∞, for all w ∈ C∞0 (Ω),

since eVkj converges strongly in L2(Ω) and ∇vkl weakly in L2(Ω), so that the product converges

weakly in L1(Ω). Using the density of C∞0 (Ω) in H1(Ω) (Thm. 4.2.1), we see that the aboveequalities also hold for every v ∈ H1

0 (Ω). Thus we have shown, using also continuity of the trace,that v = S(v, σ) and

S(vkj , σkj ) = vkj → v = S(v, σ) in L2(Ω).

NONLINEAR PDES 39

Again, uniqueness of the limit implies convergence of the whole sequence (Thm 5.1.3) and thusyields the continuity of S. The compactness of the embedding H1(Ω) → L2(Ω) together with theuniform estimate (44) yields the compactness of S.Step 4: A priori estimates. Let v ∈ H1(Ω) be a fixed point of S(·, σ). The definition of M implies

M ≥ e−ψg = vD and thus (v −M)+ = 0 on ∂Ω, so that (v −M)+ ∈ H10 (Ω) is an admissible test

function for the first equation of (34):

0 =

∫ΩeV∇v · ∇(v −M)+ dx =

∫ΩeV∣∣∇(v −M)+

∣∣2 dx,

i.e. (v−M)+ = 0 and v ≤M in Ω. Similarly, since 1/M ≤ e−ψg, taking the test function (v−1/M)−

implies (v − 1/M)− = 0 and thus v ≥ 1/M a.e. in Ω. This shows vM = v. Furthermore, we obtainuniform L∞- and L2-estimates for all fixed points of S(·, σ).Step 4: Existence of a fixed point. As all assumptions are satisfied, we can apply the Leray-Schauderfixed point theorem (Thm. 5.2.2) to conclude the existence of a fixed point of S(·, 1). Then, (v, V ),with V defined as solution to the Poisson equation (40), satisfies the boundary value problem(39)–(41). As 1/M ≤ v ≤M , (u = eV v, V ) solves the original problem (34)–(36).

If we assume more regularity on the boundary data, we have the following result

Proposition 6.2.3. Let the assumptions of Theorem 6.2.1 hold. In addition, assume that ∂Ω ∈C∞, f, g, ψ ∈ C∞(Ω). Then every weak solution to (34)–(36) is a classical solution (i.e. satisfiesthe equation and the boundary conditions pointwise everywhere).

Proof. The proof is based on the so-called bootstrapping technique: A weak solution to (35) satisfies

∆V = u− f(x) ∈ L∞(Ω),

Theorem 5.1.7 thus implies V ∈ W 2,p(Ω) for every 1 ≤ p < ∞. In implies in particular |∇V | ∈L∞(Ω) and thus, by (34)

∆u = div(u∇V ) = ∇u · ∇V + u∆V ∈ L2(Ω).

Again Theorem 5.1.7 implies u ∈ H2(Ω). Furthermore, for all i = 1, . . . , d we have

∆∂iV = ∂iu− ∂if ∈ L2(Ω),

and one more application of Theorem 5.1.7 yields ∂iV ∈ H2(Ω), i.e. V ∈ H3(Ω). Furthermore, for

1 ≤ p <∞ s.t. 1/q + 1/p = 1 for some q ∈ N∗ and using |ab|2 ≤ 2(

1p2a

2p + 1q2 b

2q)

yields

∆∂iu = ∇∂iu︸︷︷︸∈L2(Ω)

· ∇V︸︷︷︸∈L∞(Ω)

+ ∇u︸︷︷︸∈H1(Ω)

· ∇∂iV︸︷︷︸∈Lp(Ω)

+ ∂iu︸︷︷︸∈H1(Ω)

∆V︸︷︷︸∈Lp(Ω)

+ u︸︷︷︸∈L∞(Ω)

∆∂iV︸︷︷︸L2(Ω)

∈ L2(Ω),

with boundary conditions ∂iu = ∂ig on ∂Ω and thus ∂iu ∈ H2(Ω) or u ∈ H3(Ω). We can continuethis argument until we obtain u, V ∈ Hm(Ω) with m so large that (by Sobolev embedding) weconclude u, V ∈ C2(Ω).

Remark 6.2.4. In general, one cannot expect uniqueness for the drift diffusion system. There areeven physical semiconductor devices (thyristor) which make use of this non-uniqueness to be ableto be in two different operational states (hysteresis).


7. Time dependent equations - Some examples

Up to now, we have only considered stationary equations, i.e. we were looking for solutions u = u(x)depending only on the spatial variable. Let us consider some examples for time dependent nonlinearequations / systems.

7.1. Intro. The prototypical equation is the heat equation. For fixed T > 0, we are looking foru = u(x, t) solution to

∂tu = ∆u in Ω× (0, T ),(45)

with boundary conditions

u = g on ∂Ω× (0, T ) or ∂nu = h on ∂Ω× (0, T ).(46)

For time dependent equation, we also have to prescribe an initial datum

u(x, 0) = u0(x) a.e. in Ω.

Note that the heat equation is a linear equation that can be shown, for every u0 ∈ L2(Ω) to havea unique solution u ∈W := L2(0, T ;H1(Ω)) ∩H1(0, T ; (H1)∗(Ω)) where for some Banach space Bwe define the Bochner spaces as the spaces of function u : [0, T ] → B s.t. the following norms arefinite

‖u‖Lp(0,T ;B) =

(∫ T

0‖u(t)‖pBdt

)1/p

<∞ for 1 ≤ p <∞ and

‖u‖L∞(0,T ;B) = ess sup0<t<T

‖u(t)‖B <∞

Solutions to the heat equation have two important features:

(1) They exist globally in time (i.e. for t ∈ [0,∞)),(2) For every positive time, u(x, t) has global support (i.e. suppu = Ω).

We now introduce two examples of nonlinear equations that behave differently.

7.2. Porous medium equation - solutions with compact support. We consider the porousmedium equation (PME) which reads as

∂tu = ∆um for m > 1, x ∈ Rd, t > 0(47)

We can consider this equation as a quasilinear parabolic equation of the form

∂tu = ∇ · (D(u)∇u) with D(u) = mum−1.

Clearly, in points where u is zero, the equation is no longer parabolic (i.e. ∇·(D(u)∇u) is no longeran elliptic operator). Thus the PME is called degenerate parabolic equation. This also becomesimmanent when we consider the case m = 2 in which we can write

∂tu = 2u∆u+ 2|∇u|2.

If u becomes small, the equation approximates ∂tu ≈ 2|∇u|2, the Eikonal equation, a first orderequation of Hamiliton-Jacobi type.Many more details can be found in [Vaz07].

NONLINEAR PDES 41

7.3. Chemotaxis - finite time blow-up. We are interested in the evolution of a density ofbacteria ρ = ρ(x, t) and of a chemical substance called chemoattractant. We assume that thebacteria want to move to regions where the concentration of chemoattractant is high, but are alsoable to produce is themselves. This is e.g. the case, when they found a food source and want toattract more bacteria towards it. We consider Ω = R2 and prescribe the model

∂tρ = ∇ · (∇ρ− χρ∇C),

−∆c = ρx ∈ R2, t > 0(48)

with chemotactic sensitivity χ > 0 and

ρ(x, 0) = ρ0 ≥ 0.

This system features an interesting dichotomy: If the initial mass

M :=

∫R2

ρ0 dx

is below a certain threshold, namely for M < 8π/χ, solutions exist globally. For M > 8π/χ, thereexists a finite time T ∗ < ∞, such that the solutions blow up (become sum of Dirac measures) ast → T ∗ (The critical case M = 8π/χ is treated separately - here, blow-up can appear only atinfinite time see [BKLN06]). Proof is, e.g., based on considering 2nd moments and showing

d

dt

∫R2

|x|2ρ(x, t)dx = 4M

(1− χM

8π

)Blow up phenomena also occur in simpler equations, e.g.

∂tu = ∆u+ u2.

For more details on Chemotaxis, see [Per07].

7.4. Reaction diffusion equations. Finally, we consider a reaction-diffusion equation, the so-called FitzHughNagumo equation, as an example of pattern formation.On Ω = (0, 1)x(0, 1) the equation is given by

∂tu− d2u∆u− u+ u3 + v = κ,

τvt − d2v ∆v − u+ v = 0.

with positive constants du, dv and κ and with homogeneous Neumann boundary conditions andinitial conditions

u(x, 0) = u0(x) a.e. in Ω.

As is well known, due to an instability mechanism, solutions to this equation form distinct patterns.


References

[AF03] Robert A. Adams and John J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics(Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 2003.

[Alt16] Hans Wilhelm Alt. Linear functional analysis. Universitext. Springer-Verlag London, Ltd., London, 2016.An application-oriented introduction, Translated from the German edition by Robert Nurnberg.

[BKLN06] Piotr Biler, Grzegorz Karch, Philippe Laurenot, and Tadeusz Nadzieja. The 8-problem for radially sym-metric solutions of a chemotaxis model in the plane. Mathematical Methods in the Applied Sciences,29(13):1563–1583, 2006.

[Bre11] Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer,New York, 2011.

[Eva98] Lawrence C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. Amer-ican Mathematical Society, Providence, RI, 1998.

[Giu84] Enrico Giusti. Minimal surfaces and functions of bounded variation, volume 80 of Monographs in Mathe-matics. Birkhauser Verlag, Basel, 1984.

[GT01] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics inMathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[Hat02] A. Hatcher. Algebraic Topology. Algebraic Topology. Cambridge University Press, 2002.[Jue18] Ansgar Juengel. Nichtlineare partielle differentialgleichungen (lecture notes), June 2018.

http://www.asc.tuwien.ac.at/ juengel/scripts/nPDE.pdf.[Nie10] Barbara Niethammer. Fixed point methods for nonlinear pdes. online lecture notes, 2010.

https://courses.maths.ox.ac.uk/node/download material/15278.[Per07] Benoıt Perthame. Transport equations in biology. Frontiers in Mathematics. Birkhauser Verlag, Basel,

2007.[Smi14] Zachary Smith. Fixed point methods in nonlinear analysis, August 2014.

http://math.uchicago.edu/ may/REU2014/REUPapers/Smith,Z.pdf.[Tro13] Giovanni Maria Troianiello. Elliptic differential equations and obstacle problems. Springer Science & Busi-

ness Media, 2013.[Vaz07] J.L. Vazquez. The porous medium equation: mathematical theory. Oxford mathematical monographs.

Clarendon, 2007.[Zei90] Eberhard Zeidler. Nonlinear functional analysis and its applications. II/A. Springer-Verlag, New York,

1990. Linear monotone operators, Translated from the German by the author and Leo F. Boron.

Email address: [email protected]

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INTRODUCTION TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS ... · 7. Time dependent equations - Some examples 40 7.1. Intro 40 7.2. Porous medium equation - solutions with compact support