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Lecture notes: Applied Mathematics
(Variational calculus, PDEs)
Dietmar Ölz∗
Universidad Nacional de San Luis (UNSL), 2007
1 Introduction
This course in Applied Mathematics centres around the understanding of Variational Cal-culus and the application of the variational principle in PDE-theory. It tries to make acompromise between intuitional understanding and mathematical rigidity. A basic idea ofthe course is to elaborate more and more on some models of rigid bars, rst in stationarysettings, later on also in a dynamic setting.
In terms of literature, I refer to [Eva98], [SS04], [AGS05], [Amb04], [Tes04] and [GF65].
San Luis, August 2007
Contents
1 Introduction 1
2 Calculus in Banach spaces 22.1 Gâteaux-dierential, 1. Variation, directional derivative . . . . . . . . . . . 2
2.1.1 2. Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Fréchet derivative, functional derivative, tangential mapping . . . . . . . . . 32.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Variational calculus 63.1 Variational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Existence of minimisers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Convexity-uniqueness of the minimiser . . . . . . . . . . . . . . . . . . . . . 93.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4.1 Some fundamentals of Dierential Geometry . . . . . . . . . . . . . . 103.4.2 Admissible variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.3 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
∗Universität Wien, [email protected]
1
4 Gradient ows/steepest descent ows on manifolds 144.1 The curve straightening ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.2 Construction of solutions . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Nonlinear heat equations as gradient ows . . . . . . . . . . . . . . . . . . . 20
A Preliminaries from functional analysis (cp. [Eva98], [Eva90]) 21
2 Calculus in Banach spaces
Notation: Let X, Y , Z be Banach-spaces. We denote by C(X,Y ) the set of continuousfunctions from X to Y and by L(X,Y ) ⊂ C(X,Y ) the set of bounded, linear functions.
2.1 Gâteaux-dierential, 1. Variation, directional derivative
René Gâteaux (1889-1914). Let f ∈ Ω ⊂ X and U : Ω 7→ Y a function, δf ∈ X
δU(f, δf) = limε→0
U [f + εδf ]− U [f ]ε
=d
dεU [f + εδf ]
∣∣∣ε=0
Properties
• homogeneous: δU(f, kδf) = kδU(f, δf) for k ∈ R.
• in general non-linear, see examples
A function U is Gâteaux-dierentiable if the Gâteaux-dierential exists in all directions.
2.1.1 2. Variation
δ2U(f, δf)δf =d2
dε2U [f + εδf ]
∣∣∣ε=0
2.1.2 Examples
1.
U [(f1, f2)] :=
f1f2
2
f21 +f2
2
0 f1 = f2 = 0
we have δU(0, δf) = U(δf) and δU(0, (0, 1))+δU(0, (1, 0)) = 0+0 6= 12 = δU(0, (1, 1)).
2.
U [(f1, f2)] :=
f21
(1 + 1
f2
)0 if f2 = 0
⇒ δU(f, δf) :=
2f1
(1 + 1
f2
)δf1 −
f21
f22δf2
(δf1)2
δf2if f1 = f2 = 0
undened if f2 = 0 , f1 6= 0
and
δ2U(f, δf) :=
2(1 + 1
f2
)(δf1)2 − 4f1
1f22δf1δf2 + 2f2
1
f32(δf2)3
2(δf1)2 if f1 = f2 = 0
3. U(f) = ‖f‖p: δU(0, δf) = ‖δf‖ if p = 1 and δU(0, δf) = 0 if p > 1.
2
2.1.3 Application
2.2 Fréchet derivative, functional derivative, tangential mapping
Maurice Fréchet (1878-1973). A function U : Ω 7→ Y is called Fréchet-dierentiable atf ∈ Ω if there exists a bounded linear operator dU(f) : X 7→ Y such that
lim‖δf‖→0
U(f + δf)− U(f)− dU(f)δf‖δf‖
= 0 , (1)
or, equivalently,U(f + δf) = U(f) + dU(f)δf + o(δf) .
In the sequel we (also) use the notation δUδf [f ]δf := dU(f)δf .
If f : Ω 7→ R we call f ∈ Ω∗ a functional and δfδx the functional derivative. Furthermore,
if the mapping f 7→ dU(f) is continuous, we call U continuously dierentiable, U ∈C1(Ω, Y ).
One might iterate dierentiation and writing U ∈ Cr(Ω, Y ) for r = 1, 2, ... if the r-thderivative of U , drU (i.e., the derivative of the (r − 1)-th derivative of U), exists and iscontinuous. Finally, we dene C∞(U, Y ) =
⋂r∈N Cr(Ω, Y ) and, for notational convenience,
C0(Ω, Y ) = C(Ω, Y ) and d0U = U . n fact, we have
U : Ω ⊂ X 7→ YdU : Ω ⊂ X 7→ L(X,Y )d2U : Ω ⊂ X 7→ L(X,L(X,Y )) ∼= L2(X,X;Y )d3U : Ω ⊂ X 7→ L(X,L2(X,X;Y )) ∼= L3(X,X,X;Y )
...
where Ln(X, ...X︸ ︷︷ ︸n times
;Y ) denotes the spaces of multilinear mappings from Xn to Y for n =
1, 2, ....
Lemma 1. (Properties)
1. The linear mapping dU(f) is unique. (Take the dierence of (1) for two dierent
linear mappings dU and dU)
2. We have d(λU1 + µU2)(f)δf = λdU1(f)δf + µdU2(f)δf
3. (Chain rule) Let F ∈ Cr(X,Y ) and G ∈ Cr(Y, Z), r ≥ 1. Then G F ∈ Cr(X,Z)and d(G F )(x) = dG(F (x)) dF (x), x ∈ X.
4. (Taylor expansion) For U ∈ Cn(X,R), f,∆f ∈ X we have
U(f+∆f) = U(f)+dU(f)∆f+12d2U(f)(∆f,∆f)+ ....+
1n!dnU(f)(∆f, ...,∆f)+R
with R ∈ X, R = o(‖∆f‖n).
5. If U ∈ Cn+1(X), the error term can be written as R = 1(n+1)!d
n+1U(f)(∆f, ...,∆f)for af = f + λ∆f with 0 ≤ λ ≤ 1.
Proof. 3. ...like in real analysis
3
2.2.1 Examples
1. δ(f)δf · δg =
2. δ(F (f))δf · δg =, f : U 7→ R, F : R 7→ R
3. δ(F1(f)F2(f))δf · δg =
4.δ
“F1(f)
f
”δf · δg = (for f 6= 0)
5. F : Rm 7→ Rn, dF (x)δx =?
6. U : Rm 7→ R, dU(x)δx =?, δ2U(x, δx) =?, d2U(x)(δx1, δx2) =?
7. δ(f ′)δf · δg =
8. δ|f |δf · δg =, f : Ω 7→ Rn
9. δUδf [f ]δg = with U [f ] :=
∫fp + (f ′)p, f ∈W 1,p(U,Rn).
10. U(w) :=∫U L(Dw(x), w(x), x) dx with w : Rn 7→ R and L a C1 function. δU
δw (w)·δw =?.
Curves in R2 A curve is a mapping z : [0, L] ⊂ R 7→ R2. If |z′| ≡ 1 the curve isparametrised by its arc length. Its length is L and it satises
d
ds
(12|z′|2
)= z′ · z′′ = 0 .
Hence z′′ = z⊥ · z′′z⊥. If z = (x, y)t we use the notation z⊥ = (−y, x)t. Furthermorewe dene (neglecting the problem of periodicity of angles!): arg z := arctan y
x and theindicatrix ω := arg z′.
1. U [z] =∫ L0 |z′| ds, δU(z)δz =?
2. ω′ =?, show that(
z′
|z′|
)′= ω′ z
′⊥
|z′| ,(
z′⊥
|z′|
)′=?
3. U [z] =∫ L0
12ω
2 ds, δU(z)δz =?
4. U [z] =∫ L0
12 |z|
2(arg z − φ)2 ds, δU(z)δz =?
5. z : [0, 1] 7→ R2
(a) U =∫ 10
12 |z
′′|2 ds (Kirchho Bending energy), δU(z)δz =?
δU [z]δz =∫ 1
0z′′ · δz′′ ds =
∫ 1
0z′′ · z′︸ ︷︷ ︸
=0
z′ · δz′′ + z′′ · z′⊥ z′⊥ · δz′′ ds ,
(b) Up =∫ 10
12
(z′′·z′⊥|z′|p
)2ds =
∫ 10
12
(ω′
|z′|p−2
)2ds, δU2(z)δz =? (use the fact that
δ(ω′)(z)δz = (δω(z)δz)′)
4
(c) Rewrite both variations in such a way that they act as linear mappings on δz′.
δUp =∫ 1
0
(ω′
|z′|p−2
)((δω)′
|z′|p−2− (p− 2)
ω′
|z′|pz′ · δz′
)ds =
=∫ 1
0
(ω′
|z′|p−2
)((z′⊥ · δz′
|z′|2
)′ 1|z′|p−2
− (p− 2)ω′
|z′|pz′ · δz′
)ds =
=∫ 1
0
(ω′
|z′|2(p−2)
)(z′⊥ · δz′
|z′|2
)′− (p− 2)
(ω′2
|z′|2(p−1)
)z′ · δz′ ds =
=[(
ω′
|z′|2(p−2)
)(z′⊥ · δz′
|z′|2
)]1
0
−∫ 1
0
(ω′
|z′|2(p−2)
)′(z′⊥ · δz′|z′|2
)+(p−2)
(ω′2
|z′|2(p−1)
)z′·δz′ ds
(d) Evaluate them for those z which satisfy |z′| ≡ 1.
δUp[z] =∫ 1
0ω′((
z′⊥ · δz′)′− (p− 2) ω′z′ · δz′
)ds =
=[ω′z′⊥ · δz′
]10+∫ 1
0−ω′′z′⊥ · δz′ − (p− 2) ω′2z′ · δz′ ds
(e) For which p does it hold that δUδz [z] ·δz = δUp
δz [z] ·δz for all z that satisfy |z′| ≡ 1?
δU [z]δz =∫ 1
0ω′ z′⊥ · δz′′ ds =
[ω′ z′⊥ · δz′
]10−∫ 1
0
(ω′ z′⊥
)′· δz′ ds =
=[ω′ z′⊥ · δz′
]10−∫ 1
0
(ω′′ z′⊥ − ω′2 z′
)· δz′ ds = δUp=1[z]δz
(f) For p = 5/2 the variation of the bending energy is invariant with respect tovariations that respect that only act as reparametrisations, and maybe instan-taneous polymerisation and depolymerisation a the same rate, i.e. δz = z′φwith φ(0) = 0, φ(1) = 1. It turns out that
δU5/2[z]z′φ = 0 .
for all testfunctions φ with φ(0) = φ(1) = 0. Solution:
δUp · δz =∫ 1
0
(ω′
|z′|p−2
)((δω)′
|z′|p−2− (p− 2)
ω′
|z′|pz′ · δz′
)ds =
=∫ 1
0
(ω′
|z′|p−2
)((z′⊥ · δz′
|z′|2
)′ 1|z′|p−2
− (p− 2)ω′
|z′|pz′ · δz′
)ds
Hence, using (z′φ)′ = z′′φ+ z′φ′,
δUp·z′φ =∫ 1
0
(ω′
|z′|p−2
)((z′⊥ · z′′φ|z′|2
)′ 1|z′|p−2
− (p− 2)ω′
|z′|pz′ · (z′′φ+ z′φ′)
)ds =
=∫ 1
0
(ω′
|z′|p−2
)((ω′φ)′ 1|z′|p−2
− (p− 2)ω′
|z′|pz′ · z′′φ− (p− 2)
ω′
|z′|p−2φ′)ds =
=∫ 1
0
(ω′
|z′|p−2
)((ω′φ)′ 1|z′|p−2
+(
1|z′|p−2
)′ω′φ− (p− 2)
ω′
|z′|p−2φ′)ds =
=∫ 1
0
(ω′
|z′|p−2
)((1
|z′|p−2ω′φ
)′− (p− 2)
1|z′|p−2
ω′φ′)ds
5
If p = 5/2 this implies
δU5/2 · z′φ =∫ 1
0
(ω′
|z′|1/2
)((1
|z′|1/2ω′φ
)′− 1
21
|z′|1/2ω′φ′
)ds =
=∫ 1
0
(ω′
|z′|1/2
)((1
|z′|1/2ω′)′φ+
12
1|z′|1/2
ω′φ′)ds =
=∫ 1
0
(ω′
|z′|1/2
)(1
|z′|1/2ω′)′φ+
12
(ω′
|z′|1/2
)2
φ′ ds =12
(ω′
|z′|1/2
)2
φ
∣∣∣∣10
= 0 .
3 Variational calculus
Joseph-Louis Lagrange (∼1800)
3.1 Variational equations
Theorem 1. A necessary condition for a value f0 ∈ Ω ⊂ X, Ω an open set to be a (local)
minimum of the functional U among the functions in Ω is that the directional derivatives
vanish in the directions, which are admissible and in which the directional (Gâteaux) deriva-
tive exists. The set of admissible variations in this case is given by X since (f0 +ε0δf) ∈ Ωfor all 0 < ε ≤ ε0 with ε0 small enough. Hence the necessary condition reads U(f0, δf) = 0for all δf ∈ X.
Proof. Since f0 is a local minimum in Ω we have U(f0 + εδf) ≥ U(f0) for all 0 < ε ≤ ε0with ε0 small enough.
Theorem 2. (Fundamental Lemma of the Calculus of Variations). Let U be an open
subset of Rm and let f ∈ C(U,R) and let C∞c (U,R) the space of real valued C∞ functions
on U with compact support in U , i.e. test functions, then∫Uf φ dx = 0 for all φ ∈ C∞
c (U,R)
then f ≡ 0 on U .
Proof. Assume that f(x0) > 0 for x0 ∈ U , then f is also large then zero in a neighbourhoodof x0. Choose a testfunction φ > 0 with φ(x0) > 0 and support in this neighbourhood andconclude the contradiction.
The same statement for f ∈ L2(U) is a consequence of Lebesgue's Theorem, cp. [H03].
We apply this statement in the following way: For a minimisation problem in thesetting of Theorem 1, if C∞
c (U) ⊂ X for some open set U ⊂ Rn, we might assume the thederivatives are in fact classical and apply integration by parts. Applying Theorem 2 we thenderive the strong formulation of the pde which we might seek to solve explicitly. Solutionsof this strong formulation are also solutions of the weak formulation (by bootstrapping!),but not vice versa.
Problem 1. A model for a sti bar under load: h : [0, L] 7→ R, h(0) = 0, h′(0) = 0.Admissible variations A = δh ∈ H2([0, L],R), δh(0) = 0, δh′(0) = 0. Minimise U [h] =∫ 10 κ/2 |h
′′|2 ds + h(L)M where L > 0 is the length of a sti bar, M > 0 refers to a massattached to the tip and κ > 0 to the stiness.
6
Let h a minimiser, then Theorem 1 implies
dU(h)δh =∫ L
0κh′′δh′′ + δhM = 0 for all δh ∈ X := A
Assume h has classical derivatives up to fourth order and specialise δh ∈ C2((0, L),R) withδh(0) = δh′(0) = 0. Integration by parts yields
0 = κh′′(1)δh′(1)− κh′′′(1)δh(1) +∫ L
0κh′′′′δh+ δhM
We concludeh′′(1) = 0 , κh′′′(1) = M (2)
and specialising δh ∈ C∞c ((0, L)) and applying Theorem 2 we conclude
h′′′′ = 0 on (0, L) (3)
The system (2), (3) allows to compute the solution h(s) = MLκ s2
(s/L6 − 1
2
).
Example 1. Consider the minimisation problem U [x] =∫V L(x(y), Dx(y), y) dy among
all x in the appropriate function space X such that U [x] exists and where we restrict tothose admissible functions which satisfy x∂V = g on the boundary of V ⊂ R bounded, fora suciently regular function g that lives on the boundary. This is a typical setting intheoretical mechanics, the function L is called Lagrangian. The weak formulation of thevariational equation reads
0 =δU
δx[x] · v =
∫V
∂L
∂xv +
∂L
∂x′Dv dy
for all v ∈ X with v|∂V = 0 and we get the strong formulation from
0 =∫
V
∂L
∂xv +
∂L
∂x′Dv dy =
∫V
∂L
∂xv − divy
(∂L
∂x′
)v dy
and hence ∂L
∂x− divy
(∂L
∂x′
)= 0 on V ,
x|∂V = g .
is the strong formulation of the Euler-Lagrange equation.
3.2 Existence of minimisers
We use the example:
Problem 2. Minimise
U [z] =∫ 1
0
κ
2|z′′|2 ds+ z2(1)M with z =
(z1z2
)(4)
on the set of admissible functions A
A =w ∈ H2((0, 1),R2), |w′| ≡ 1, w(0) =
(00
), w′(0) =
(10
).
The constant M > 0 refers to a mass, attached to the endpoint of the curve.
7
Observe that due to Morrey's inequality we have w ∈ A ⇒ w ∈ C1, 12 ([0, 1],R2), which
justies the evaluation of w(1), w(0) and w′(0) in (4) and in the denition of A and whichimplies that the set A ⊂ H2((0, 1),R2) is closed with respect to weak convergence in H2.
Lemma 2. The functional (4) is coercive on the set of admissible functions A in the sense
that
U [w] ≥ C1‖w‖2H2((0,1), R2) − C2 ∀ w ∈ A
for some constants C1 > 0 and C2 > 0.
Proof. Observe that ∫ 1
0|w′′|2 ds ≤ 2
κU [w] +
2κM
where we used that z2(1) ≥ −1. ∫ 1
0|w′|2 ds = 1 .
The fact that |w(s)| ≤ 1 concludes the proof.
Lemma 3. The functional (4) is weakly lower semicontinuous in the sense that
U [w] ≤ lim infk→∞
U [wk] whenever wk w weakly in H2((0, 1),R2) .
Proof. The proof is modeled after [Eva98], p. 446 and [LU68] respectively and uses theconvexity of the integrand with respect to the highest order derivatives (cp. [Str00], p. 12,remark 1.8).
Assume the existence of a subsequence such that U [w] > limk→∞ U [wk]. The functional∫ 10
κ2 |z
′′|2 ds may be expanded as∫ 1
0
κ
2|w′′k |2 ds =
∫ 1
0
κ
2|w′′|2 ds+
∫ 1
0κw′′ · (w′′k − w′′) ds+
12
∫ 1
0κ|w′′|2 ds
with a mean value w = λw + (1− λ)wk, 0 < λ ≤ 1. Hence we infer the inequality
U [wk] ≥∫ 1
0
(κ2|w′′|2 + κw′′ · (w′′k − w′′)
)ds+ (wk)2(1)M
due to the apparent convexity of∫ 10
κ2 |z
′′|2 ds. Boundedness in H2((0, 1),R2) allows topick a subsequence which is strongly convergent in H1((0, 1),R2) and also convergent inC((0, 1),R2). This implies
limk→∞
U [wk] ≥∫ 1
0
κ
2|w′′|2 ds+ w2(1)M = U [w] .
contradicting the assumption.
Theorem 3. There exists a minimiser of the energy functional (4) in the set of admissible
functions A.
Proof. Letm := inf
w∈AU [w]
and let (wk)k∈N be a minimising sequence, i.e.
wk ∈ A and U [wk] → m as k →∞ .
8
This together with Lemma 2 implies that there is a function w such that for a subsequencewe have
wkj w weakly in H2((0, 1),R2) .
We conclude that wkj→ w in H1((0, 1),R2) and therefore
1 = |w′kj| → |w′| a.e on (0, 1) .
Since we have |w′k| ≡ 1 for all k and since w ∈ C1,1/2([0, 1],R2) we have |w′| ≡ 1, i.e. theweak limit w lies in the set of admissible functions A.
As U is weakly lower semi continuous by Lemma 3, the function w is a minimiser of(4) under the constraint |w′| = 1.
Remark 1. Morally we have:
Coercivity+weak lower semicontinuity+weak closedness of the set of admissible functions⇒ Existence of a minimiser
3.3 Convexity-uniqueness of the minimiser
Denition 1. A function U ∈ C2(Ω,R), Ω ⊂ X an open subset, is convex at a point f0 ∈ Ωif d2U(f0)(δf, δf) ≥ 0 for all δf ∈ X. It is strictly convex if d2U(f0)(δf, δf) ≥ ϑ‖δf‖2 fora constant ϑ > 0.
The functional U is uniformly convex in Ω if d2U(f)(δf, δf) ≥ ϑ‖δf‖2 for all f ∈ Ωand all δf ∈ X and a constant ϑ > 0.
Theorem 4. A necessary condition for a functional U ∈ C2(Ω,R), Ω ⊂ X an open subset
to have a local minimum at a critical point f0 ∈ Ω is that it is convex at f0.
Proof. Assume that there is δf ∈ X such that d2U(f0)(δf, δf) < 0 with ‖δf‖ = 1. Hencethere is a ball B ⊂ Ω such that d2U(f)(δf, δf) < C < 0 for all f ∈ B. Let 2r be the radiusof the ball B, then we infer
U(f0 + rδf) = U(f0) + rdU(f0)δf +12d2U(f)(δf, δf)
< U(f0)
since dU(f0) = 0 and the mean value f = f + λδf (with 0 ≤ λ ≤ 1) is in B. Thiscontradicts the assumption.
Theorem 5. A sucient condition for a functional U ∈ C2(Ω,R), Ω ⊂ X an open subset
to have a local minimum at a critical point f0 ∈ B ⊂ Ω, B ⊂ Ω an open ball, which is
unique in B, is that it is uniformly convex in B.
Proof. Assume that there is a f1 ∈ B with U(f1) ≤ U(f0) and set δf = f1 − f0. We infer
U(f1) = U(f0 + δf) = U(f0) + dU(f0)δf +12d2U(f)(δf, δf)
≥ U(f0) +12ϑ‖δf‖2
since the mean value f is in B and using dU(f0) = 0. This contradicts the assumption.
9
Example 2. The functional U [h] =∫ L0
κ2h
′′2 ds + h(L)M from Problem 1 is globallyconvex. Using h(0) = h′(0) = 0 we infer∫ L
0|h′|2 ≤ L2
2
∫ L
0|h′′|2 and
∫ L
0|h|2 ≤ L2
2
∫ L
0|h′|2 ≤ L4
4
∫ L
0|h′′|2 ,
which implies
‖δh‖2H2 ≤
(1 +
L2
2+L4
4
)∫ L
0h′′2 ds .
Therefore we have d2U(h)(δh, δh) =∫ L0 κδh′′2 ds ≥ ϑ‖δh‖2
H2 for ϑ = κ
1+L2
2+L4
4
.
3.4 Constraints
3.4.1 Some fundamentals of Dierential Geometry
Let φ : U ⊂ X 7→ V ⊂ M ⊂ Y be an invertible map such that φ ∈ C∞(U, Y ). LetV := φ(U) ∩ M and denote the inverse by ψ := φ−1 : V 7→ U . We call φ a localparametrisation of the manifold M and ψ a local map of M .
Let ψii∈I a family of maps to the manifold M with ψi : Vi ⊂ M 7→ ψ(Vi) ⊂ X andψi ∈ C∞ invertible. If
⋃i∈I Vi = M we call this family an atlas of the manifold M .
Let f ∈ Ω ⊂ X, x := φ(f) ∈M and consider the tangential mapping Tφ(f) := dφ(f) :X 7→ TM(x) ⊂ Y . The notation TM(x) (tangent space to M at x ∈M) is motivated bythe fact that TM(x) is a (sub)vectorspace of X. Its vector space structure is taken fromX via dφ(x).
nite dimensions: curves, manifolds, tangent space, linear functionals (dual space),tensors, ows on a manifold
Note: the theory of Dierential Geometry is fully developed in the case when thevector spaces used in the above denitions are Rn with n ∈ N. When we are talking aboutfunction spaces instead, a consistent formulation of the theory might not be available.
3.4.2 Admissible variations
Let U : Y 7→ R be a functional and consider the variational problem of minimising itamong the set of admissible functions A ⊂ Y . Assume that the set A can be described asa manifold M = A, covered by the images of a family of parametrisations φii∈I .
Going back to the functional U φi : Ui 7→ R, we observe that we are again in thesituation of Theorem 1, i.e. the set of admissible functions in this case is the open setUi ⊂ X and the admissible variations are given by the function space X. The necessarycondition for f0 ∈ Ui being a local minimum of U φi according to Theorem 1 readsd(U φi)(f0)δf = 0 for all δf ∈ X. We get
d(U φi)(f0)δf = 0 = dU(φi(f0)) dφi(f0) δf
We denote x0 = φi(f0) and δx := dφi(f0) δf . This suggests, for the minimisationproblem of minimising U with A, to dene the set of admissible variations at x0 ∈ M byBx0 := δx ∈ Y : δx = dφi(ψ(x0))δf, δf ∈ X. Hence Bx0 = TM(x0), the tangent spaceof the manifold M at x0 ∈M .
Assume that the set of admissible functions A is being described as the set of solutionsto an equation: f ∈ Y : G(f) = 0. Hence we have G φi(f) = 0 for all f ∈ Ui and,since the linearisation of a constant function is zero, we get dG(φi(f)) dφi(f)δf = 0 for all
10
δf . This implies that the admissible variations δx are those which are in the kernel of thelinearised equation,
dG(f) · δx = 0 .
This is intuitively clear in the sense that those variations are admissible, in the directionsof which the constricting object G does (asymptotically) not change.
It also is reected in the usual argumentation in nite dimensional optimisation, whereit is stated that minima under a constraint are found where the contour line of the func-tional (the directions in which the functional does asymptotically not change!) are orthog-onal to the gradient of the constraint.
Examples
1. Constraint z(0) = 0 ⇒ δz(0) = 0.
2. Constraint z′(0) = 0 ⇒ δz′(0) = 0.
3. Constraint |z′|2 ≡ 1, hence, at every point s: 2z′ · δz′ = 0
4. For a given z ∈ H2([0, 1],R2) we consider the set of reparametrisation functionsX := ψ ∈ H2([0, 1], [0, 1]) : ψ(0) = 0, ψ(1) = 1 and its (open) subset Ω = ψ ∈X : ψ′(s) > 0 ∀s ∈ [0, 1]. Then A := w ∈ H2([0, 1],R2) : w = z ψ, ψ ∈ Ω,i.e. the set of all C∞ reparametrisations of the curve z. The parametrisation of thisadmissible set is φ : Ω 7→ A, ψ 7→ z ψ. Variations, which are admissible with respectto A are δz = dφ(id)dψ = z′ id dψ = z′dψ with dψ ∈ X.
The existence of an equivalent denition of A using condition G(z) = 0 is, in thecase of nite dimensions, showed using the implicit and inverse function theorems.
5.
Problem 3. Minimise U(z) =∫ 10
κ2 |z
′′|2 + z2(1)M within the set of admissible func-tions A = w ∈ H2((0, 1),R2) : w(0) = (0, 0), w′(0) = (1, 0), |w′| ≡ 1: Parametrisa-
tion φ : X 7→ A, φ(ω) =∫ s0
(cos(ω)sin(ω)
)ds with X := ω ∈ H1((0, 1)) : ω(0) = 0. We
choose δω ∈ X. The admissible variations are given by δz(z) = dφ(ω = φ−1(z))δω =∫ s0
(− sin(ω)cos(ω)
)δω ds =
∫ s0 z
′⊥δω ds.
Observe that the set of admissible variations Bz can also be characterised by δ |z′|22 =
0 = z′ · δz′, hence Bz = w ∈ H2((0, 1),R2) : δw(0) = (0, 0), δw′(0) = (0, 0), z′ · δw′ ≡0.Consider the minimisation problem in X: Minimise U := U φ(ω) =
∫ 10
κ2 |ω
′|2 +M∫ 10 sin(ω) ds. We get δU(ω)δω =
∫ 10 κω
′ δω′ + M cos(ω)δω ds = κω′(1)δω(1) +∫ 10 −κω
′′δω + M cos(ω)δω ds. This suggests to solve the variational equation (cp.gure 1)
κω′′ = M cos(ω) on (0, 1) ,ω(0) = 0 , ω′(1) = 0 .
Note that in Theorem (3) we already proved the existence of a solution to the min-imisation problem and therefore to the variational equation.
Furthermore we have d2U(ω)(δω, δω) =∫ 10
[κδω′2 −M sin(ω)δω2
]ds, which implies
that the minimisation problem is convex for all ω with −π < ω(s) ≤ 0 for s ∈ (0, 1).
11
−0,05
0,9
−0,15
0,7
s
0,0
1,0
−0,1
−0,2
0,8
−0,25
−0,3
0,60,50,40,30,20,10,0
Figure 1: Solutions of Problem 3 (line) and Problem 1 (points) for M = L = κ = 1
3.4.3 Lagrange multipliers
We could apply Theorem 2 directly to problem 1 and problem 3, since testfunctions C∞c (U),
U open in Rn were also elements of the space of admissible variations B = X. In the settingof problem 3 we can not do the same on the level of the minimisation problem minx∈A Usince testfunctions are not elements of Bx. To circumvent this problem we introduceLagrange multipliers.
If f0 ∈ A ⊂ Y is a minimiser of U [f ] in the set of admissible functions A, we know thatδUδf [f0] · δf = 0 for all admissible variations δf ∈ Bf0 . We may write Y ∼= Bf0 ⊕ Y/Bf0 . Wedene the linear functional λ[f0] ∈ Y ∗
λ[f0] : Y 7→ Rv 7→ − δU
δf [f0] · v
assuming that the variation of U exists in all directions in Y . Observe that this implies
〈λ[f0], δf〉 = 0 for all δf ∈ Bf0
andδU
δf[f0] · v + 〈λ[f0], v〉 = 0 for all v ∈ Y .
Observe that due to the denition of the Lagrange multiplier as counteracting eectwe may morally interpret it as the force that enforces the constraint. A typical exampleis the interpretation of pressure as a force which ensures incompressibility of a uid.
When seeking to compute the Lagrange multiplier, one has to make use of the infor-mation, that it vanishes on the subvectorspace B, i.e. it acts on the factor space Y/B:For a constraint given by G(u) = 0 with G ∈ C1(Y, Z), Z a Banach space, we may writethe Lagrange multiplier as 〈λ, δu〉 = 〈λ, dG(u)δu〉 implying that admissible variations arein the kernel of the Lagrange multiplier and that it is a linear functional on the image ofdG(u).
We distinguish two canonical situations (cp. [GH96]):
1. Isoparametric constraint G(0) = 0 with G : Y 7→ R. Here the Lagrange multiplierbasically acts as a linear functional on R, i.e. it is a real number and writes 〈λ, δu〉 =λdG(u)δu. The name refers to the fact that we might be minimising among geometricobjects with a xed surface or volume. The functional G then represents the surfaceor the volume or the length of the object.
12
2. (Non)holonomic constraints represent pointwise conditions on the function (holo-nomic) or derivatives of the function (nonholonomic). If the (pointwise) condi-tion reads G(u(.), u′(.), ..., .) ≡ 0, the heuristic approach is to make the ansatz〈λ, δu〉 =
∫λdG(u)δu for a scalar valued function λ.
Example 3. (Isoparametric constraint) Minimise U [u] =∫ 1−1 u
′2 dx among the set of
admissible functions A = u ∈ H10 ((−1, 1),R) :
∫ 1−1 u dx = 2, i.e. among all functions
u ∈ H1 with value zero (in a certain sense!) at the boundary and average value equal to 1.Considering the variation of the constraint G(u) =
∫u dx − 2 = 0 we obtain the
set of admissible variations B = δu ∈ H10 ((−1, 1),R) :
∫ 1−1 δu dx = 0. Observe that
C∞c ((−1, 1)) * B. Hence we look at the larger space Y = H10 ((−1, 1)) and consider the
factor space Y/B = [δu]∼ where δu1 ∼ δu2 ⇔∫δu1 =
∫δu2. We identify δu ∈ Y with∫
δu ∈ R and infer Y/B ∼= R. Linear functionals on this factor space are represented by
real values acting on∫ 1−1 δu dx for δu ∈ Y .
dU(u)δu = 2∫ 1
−1u′δu′ dx = 0 for all δu ∈ B
and the Lagrange multiplier 〈λ, v〉 = λ∫v dx. This implies
0 = 2∫ 1
−1u′δu′ dx+ λ
∫ 1
−1δu = 0 for all δu ∈ Y . (5)
Observe that constraints dened by G(u) = 0 for G ∈ C1(Y,R) are called isoperimetricconstraints. Lagrange multipliers associated to them read 〈λ, δu〉 = λdG(u)δu for a scalarvalue λ ∈ R and δu ∈ Y .
We assume that the solution of (5) is C2 and derive the system0 = −2u′′ + λ = 0 ,u(−1) = u(1) = 0 ,∫ 1
−1u dx = 2 .
which yields λ = −6 and u = λ4 (x2− 1) = 3
2(1− x2). Observe that this solutions is uniquesince d2U(u)(δu, δu) = 2
∫ 1−1 δu
′2 dx ≥ 23‖δu‖
2H1 . (Same computation as above). Observe
that B is a subspace!
Example 4. (nonholonomic constraint) A typical example would be the minimisationproblem in Problem 3 with the constraint |z′| ≡ 1 on Y := z ∈ H2([0, 1],R2) : z(0) =(0, 0), z′(0) = (1, 0).
The heuristic approach is to make the ansatz 〈λ, δz〉 = 〈λ, dG(z)δz〉 =∫ 10 λz
′ · δz′ dsfor a scalar valued function λ : [0, 1] 7→ R.
To get more insight we may formalise this as follows: We write a general variationδz ∈ Y with Y := w ∈ H2((0, 1),R2) : w(0) = w′(0) = (0, 0) as
δz =∫ s
0δz′ ds =
∫ s
0z′ z′ · δz′︸ ︷︷ ︸
=:ϑ
ds+∫ s
0z′⊥ z′⊥ · δz′︸ ︷︷ ︸
=:ω
ds
with ϑ, ω ∈ H1((0, 1),R), ϑ(0) = ω(0) = 0. Hence we have B = ∫ s0 z
′⊥ω : ω ∈ H1((0, 1))and infer Y/B ∼=
∫ s0 z
′ϑ : ϑ ∈ H1((0, 1)).
13
The Lagrange multiplier now can be seen as a linear functional on the factor spaceY/B which is parametrised by ϑ ∈ H1((0, 1)), with ϑ(0) = 0. In fact we note that∫ 10 λz
′ · δz′ =∫ 10 λϑ.
For the functional from Problem 3 we get
dU [z]δz =∫ 1
0κz′′ · δz′′ ds+ δz2(1)M
and dene the Lagrange multiplier
〈λ, δz〉 = −dU(z)δz = −∫ 1
0κz′′ · δz′′ ds− δz2(1)M
we get ∫ 1
0λz′ · δz′ ds = −dU(z)δz = −
∫ 1
0κz′′ · δz′′ ds− δz2(1)M .
Specialising δz =∫ s0 z
′ ϑ ds where ϑ as above we conclude∫ 1
0λϑ ds = −
∫ 1
0κ|z′′|2ϑ ds−M
∫ 1
0z′2ϑ ds .
where we used z′′ · z′ ≡ 0. Hence, using the fundamental Lemma 2 (for f ∈ Lp) we havethat λ = −κ|z′′|2 −Mz′2 a.e. and hence λ ∈ L1((0, 1)). The weak form of the variationalequation reads 0 =
∫ 1
0
[κz′′ · δz′′ + λz′ · δz′
]ds+ δz2(1)M
|z′| = 1 , z(0) = (0, 0) , z′(0) = (1, 0)
for all δz ∈ Y and the strong formz′′′′ − (λ1z
′)′ = 0κz′′′(1)− λ1(1)z′(1) = (0,M)z(0) = (0, 0) , z′(0) = (1, 0) , z′′(1) = (0, 0) ,|z′| = 1 .
(6)
4 Gradient ows/steepest descent ows on manifolds
4.1 The curve straightening ow
4.1.1 Introduction
We consider the gradient ow generated by the Kirchho bending energy
U [w] :=∫ 1
01/2 |w′′|2 ds (7)
on the set of open curves
A =w ∈ H2((0, 1),R2) : |w′| ≡ 1
. (8)
which restricts the function space H2((0, 1),R2) to those functions which satisfy
|w′| ≡ 1 . (9)
14
It is described by the system (the derivation follows)zt + z′′′′ − (λ1z
′)′ = 0λ1(0) = λ1(1) = 0 ,z′′(0) = z′′(1) = 0 ,|z′| = 1 ,z(t = 0, .) = zI(.) .
(10)
Observe that λ|s=0,1 = 0 is equivalent to z′′′ − λz′|s=0,1. Observe that due to Morrey'sinequality we have
w ∈ A ⇒ w ∈ C1, 12 ([0, 1],R2) , (11)
which implies that the set A ⊂ H2((0, 1),R2) is closed.The derivation of the system (10) becomes (formally) clear, when we consider the
Denition 2. Recursive scheme: Let τ > 0 be the constant size of the time steps andn = 0, 1, ... the index of the respective time step. Starting with zI ∈ A we nally dene asequence of functions Zn
τ by the recursive scheme
Z0τ = zI and Zn
τ ∈ argminw∈A Unτ [w] . (12)
where
Denition 3. Given τ , n and Zn−1τ ∈ A we dene the potential U
nτ as a modied version
of (7),U
nτ [w] := U [w] + Un
τ [w] (13)
with
Unτ [w] :=
∫ 1
0
|w(s)− Zn−1τ |2
2τds . (14)
For the formal derivation of the system (10) we consider the variational equation of theminimisation problem (13)
0 = dUnτ (Zn
τ )δz =∫ 1
0
[(Zn
τ )′′ · δz′′ + Znτ − Zn−1
τ
τδz + λn
τ (znτ )′ · δz
]ds
with the Langrange multiplier λ taken from Example 4. In the (formal) limit as τ → 0this becomes the weak (in space) formulation of the system (10).
When we restrict to admissible variations δz (see below), then the (in space) weakformulation may be written as
0 = dU(z)δz =∫ 1
0
[z′′ · δz′′ + zt δz
]ds
for all admissible variations δz. This describes an equation in the dual space of the tangentspace associated to the constraint |z′| ≡ 1 (i.e. in the co-vectorspace).
4.1.2 Construction of solutions
Theorem 6. There exists a minimiser of the energy functional (13) in the set of admissible
functions A.
Proof. Analogous to Theorem 3.
15
Denition 4. When 0 < τ < τ0 is xed, we measure the distance of the timestep Znτ and
an arbitrary lament represented by w ∈ A by
dnτ (w) :=
(∫ 1
0|w(s)− Zn
τ (s)|2 ds) 1
2
, (15)
which implies dnτ (w)2/(2τ) = Un+1
τ [w].We also dene an analogous notion of distance to measure the distance of the state of
the system at dierent points in time t2 and t1,
dτ (t1, t2) : dτ (t1, t1) = 0 andd
dt2dτ (t1, t2) = dn
τ (Zn+1τ )
1τ
(16)
where n such that nτ < t2 ≤ (n+ 1)τ .
This notion of distance allows to control the L2((0, 1)) distance.
Lemma 4. It holds that
‖Znτ (s)− Zm
τ (s)‖L2((0,1)) ≤ dτ (nτ,mτ) .
Proof. (triangle inequality!) We start from
Znτ − Zm
τ =n−1∑k=m
Zk+1τ − Zk
τ
ττ .
By applying the L2-norm on the interval (0, 1) on both sides we get
‖Znτ − Zm
τ ‖L2((0,1)) =(∫ 1
0|Zn
τ − Zmτ |2)1/2
≤∫ nτ
mτ
1τdd t
τe−1
τ
(Zd t
τe
τ
)dt = dτ (mτ, nτ) ,
Denition 5. Given the sequence (Znτ )n=1,2,... we dene the interpolating functions for
t ≥ 0 and 0 < s < 1 ,
zτ (t, s) := Znτ (s) ,
zτ (t, s) := Znτ (s)−
(n− t
τ
)(Zn
τ (s)− Zn−1τ (s)) ,
(17)
where n is such that τ(n− 1) < t ≤ τn.
Observe that
∂tz =1τ
(Zn
τ (s)− Zn−1τ (s)
)and also ‖∂tzτ (t)‖L2((0,1),R2) = 1
τ dn−1τ (Zn
τ ). Therefore∫ t1
t2
‖∂tzτ (t)‖L2((0,1),R2) dt =∫ t1
t2
1τdd t
τe−1
τ
(Zd t
τe
τ
)dt = dτ (t1, t2) .
Theorem 7. (Convergence) Let zτ (t, s) and zτ (t, s) as dened in (17), then there is a
function
z ∈ C0, 12 ([0, T ];L2((0, 1))) ∩ L∞([0, T );H2([0, 1])) ∩H1([0, T );L2((0, 1)))
and a sequence (τi)i=1,2,... with τi → 0 such that
16
1. zτi → z in Cloc([0, T ];L2((0, 1))),
2. zτi z weakly* in L∞([0, T );H2((0, 1))),
3. zτi z weakly in H1([0, T );L2((0, 1))),
Observe that here and in the following proof we do not indicate the target vector space of
the involved vector valued functions.
Proof. First we will proof the uniform convergence zτi → z and the weak convergencezτi z in L∞([0, T );H2((0, 1))) for an appropriate subsequence. In a second step we provethe weak convergence zτi z in H1([0, T );L2((0, 1))) and nally the equality z := z = z.
Fix τ > 0 and 0 < t < T . Furthermore let n such that (n − 1)τ < t ≤ nτ implyingzτ (t) = Zn
τ .Following the argumentation in [AGS05] and [Amb04] we dene the Moreau-Yosida
approximation (cp. the Yosida aproximation in the classical theory, e.g. in [Eva98])
Φnτ [Zn
τ ] := minw∈A
12τdn
τ (w)2 + U [w]
and infer
Φnτ [Zn
τ ] =12τdn
τ (Zn+1τ )2 + U [Zn+1
τ ] ≤ 12τdn
τ (Znτ )2 + U [Zn
τ ] = U [Znτ ] .
This implies12τdn
τ (Zn+1τ )2 = Φn
τ [Znτ ]− U [Zn+1
τ ] ≤ U [Znτ ]− U [Zn+1
τ ] . (18)
Summing up (18) we get
12
∫ nτ
mτ
(d
dtdτ (mτ, t)
)2
dt =n−1∑k=m
12τdk
τ (Zk+1τ )2 ≤ U [Zm
τ ] . (19)
By Jensen's inequality this implies
dτ (mτ, nτ)2 ≤ (n−m)τ∫ nτ
mτ
(d
dtdτ (mτ, t)
)2
dt ≤ 2τ(n−m)U [Zmτ ] . (20)
Together with Lemma 4 we get for two arbitrary point in times t1 > t2 ≥ 0
lim supτ→0
‖zτ (t1)− zτ (t2)‖2L2(0,1) ≤ 2(t1 − t2)Φmax ,
where Φmax = U(zI) is the maximal energy in the system, i.e. the energy of the initialcondition. An application of a rened version of the Arzelà-Ascoli-Theorem as presentedon p.69 of [AGS05] yields the result (1). This yields the Hölder-12 continuity in time of thelimit function.
Observe that as a consequence of the decay of energy (U(Zn+1τ ) ≤ U(Zn
τ )) the functionzτ is uniformly bounded in L∞([0, T );H2((0, 1))), which yields the weak∗ convergence inthis space.
Concerning (3.) we use (19) to conclude
∫ T
0
∫ 1
0|∂tzτ |2 ds dt =
∫ T
0
∫ 1
0
∣∣∣∣∣Zdt/τeτ − Z
dt/τe−1τ
τ
∣∣∣∣∣2
ds dt ≤dT/τe−1∑
k=0
dkτ (Z
k+1τ )2
τ2τ ≤ 2U(zI) .
17
This concludes the proof of point (3.).Finally we have z = z for some choice of the converging subsequences since by Lemma
4 and by (20) we have
sup0≤t≤T
‖zτ (t)− zτ (t)‖2L2((0,1)) ≤ sup
0≤t≤Tdτ
(⌈ tτ
⌉τ,(⌈ tτ
⌉− 1)τ
)2
≤ Cτ2
for some constant C > 0 by (20).
consistency We are now ready to verify the consistency of the approximation scheme(12) with the formal limit problem (10).
To this end we consider an admissible function w ∈ A with A from (8) and its general,i.e. not necessarily admissible, variation δw ∈ H2((0, 1),R2) which we write as
δw = δw(1) +∫ s
1δw′ ds =
∫ s
1
(w′ · δw′ w′ + w′⊥ · δw′ w′⊥
)ds .
Observe that w′⊥ · δw′ ∈ H1((0, 1)), w′ · δw′ ∈ H1((0, 1)) and δw(1) ∈ R2.We assume that the variation depends in an L2 sense on time (the prime ′ then always
refers to dierentiation with respect to space) and dene
ξ ∈ L2([0, T ); R2) , ω ∈ L2([0, T );H1(0, 1)) and ϑ ∈ L2([0, T );H1(0, 1)) (21)
as time depend variations of the barbed end and of the direction of the standard lamentdecomposed into the variation perpendicular to the lament and in the direction of thelament. This allows to write a L2-time dependent general variation of z ∈ A as
δw = ξ +∫ s
1
(ϑ w′ + ω w′⊥
)ds , (22)
which impliesδw′ = ϑ w′ + ω w′⊥ . (23)
Variation of the constraint (9) applied to w, i.e. |w′| ≡ 1, yields w′ · δw′ = 0, which holdsin (22) if ϑ ≡ 0. Hence an L2-time dependent admissible variation of w ∈ A is given by
δw = ξ +∫ s
1w′⊥ ω ds , (24)
with ξ and ω as in (21). For any w ∈ A we consider admissible variations δw =δw[ξ(t, .), ω(t, .)] to be linear mappings applied to the pair of functions (ξ, ω), which arethen the "real" testfunctions of our equations.
By construction, the sequence Znτ satises for all (ξ, ω) ∈ L2([0, T ); R2 ×H1(0, 1)) the
variational equations of the minimisation problem (12)(δUn
τ + δU)
[Znτ ]δZn
τ [ξ, ω] = 0 , (25)
with
δUnτ [Zn
τ ]δZnτ =
∫ 1
0
Znτ − Zn−1
τ
τ· δZn
τ ds ,
δU [Znτ ]δZn
τ =∫ 1
0(Zn
τ )′′ · (δZnτ )′′ ds .
18
Theorem 8. Let zI ∈M, then there exists a time T > 0 and a function z ∈ C0, 12 ([0, T ];L2((0, 1),R2))∩
L∞([0, T );H2([0, 1],R2)) ∩H1([0, T );L2((0, 1),R2)) such that
z(t = 0, s) = zI(s) on [0, 1] , |z′(t, s)| = 1 on [0, T )× [0, 1]
and the function z satises the system∫ T
0ξ ·∫ 1
0∂tz ds dt = 0 (WP1a)
∫ T
0
∫ 1
0z′′ · z′⊥ ω′ −
[z′⊥(s) ·
∫ s
0∂tz ds
]ω ds dt = 0 (WP1b)
for all testfunctions (ξ, ω) ∈ L2([0, T ); R2 ×H1(0, 1)).
Proof. We may plug in the approximating functions zτ and zτ .A weak-in-time version of (25) with testfunctions ξ and ω as above holds. For the
single components of this equation we get
∫ T
0δUτ [zτ ]δzτ dt =
∫ T
0
(∫ 1
0∂tzτi ds
)· ξ dt−
∫ T
0
∫ 1
0z′⊥τi
·(∫ s
0∂tzτi ds
)ω ds dt
τ→0−→
τ→0−→∫ T
0
(∫ 1
0∂tz ds
)· ξ dt−
∫ T
0
∫ 1
0
(z′⊥ ·
∫ s
0∂tz ds
)ω ds dt
where z is the common limit of Theorem 7. Analogously we get∫ T
0δU [zτ ]δzτ dt =
∫ T
0
∫ 1
0z′′τ · z′⊥τ ω′ ds dt
τ→0−→∫ T
0
∫ 1
0z′τ · z′⊥τ ω′ ds dt .
Hence the limit function z satises the system (WP1a), (WP1b).
Lemma 5. Let Znτ ∈ A be a minimiser of the functional (13) with respect to the set of
admissible functions A according to Theorem 6, then there exists a Lagrange multiplier
function λnτ ∈ L1((0, 1)) such that Zn
τ is a critical point of the functional
Unτ [w] + UL[w] with UL[w] :=
∫ 1
0λ
12|w′|2 ds (26)
and with Unτ [w] from (13) with respect to the larger set of admissible functions
z : z ∈ H2((0, 1),R2)
,
which does not take into account the constraint (9).
Proof. Consider a lament w ∈ A and a general variation δw given by (22). We split thisarbitrary variation δw into an admissible variation (24) and into a non-admissible variation
ψ(w) :=∫ s
1ϑ ∂sw ds .
Evaluating at a point Znτ we consider the variation (25) with respect to a general
variation (22) and observe that the components that refer to the admissible variation (24)cancel, whereas we are left with the nonadmissible variation ψ obtaining
δUnτ [Zn
τ ] δZnτ [ξ, ω, ϑ] =
(δUn
τ + δU)
[Znτ ] ψ(Zn
τ )[ϑ] , (27)
19
with
δUnτ [Zn
τ ]ψ(Znτ )[ϑ] = −
∫ 1
0(Zn
τ )′ ·∫ s
0
Znτ − Zn−1
τ (s)τ
ds ϑ ds ,
δU [Znτ ]ψ(Zn
τ )[ϑ] =∫ 1
0(Zn
τ )′′ · (Znτ )′′ϑ ds .
This is a linear mapping acting on ϑ. We dene λnτ as that
λnτ := |(Zn
τ )′′|2 − (Znτ )′ ·
∫ s
0
Znτ − Zn−1
τ (s)τ
ds . (28)
and infer that λnτ is bounded with respect to the L1((0, 1))-norm. Observe that higher
exponent Lp((0, 1)) estimates can not be obtained because of the |(Znτ )′′|2-term.
We dene the piecewise in t constantly interpolating function
λτ (t, s) := λdt/τeτ (s)
and conclude
Lemma 6. There is a time dependent measure λ ∈ L∞([0, T );M((0, 1))) such that λτ∗ λ
weak∗ in the space L∞([0, T );M((0, 1))).
Proof. Based on (28) we obtain a uniform in τ estimate of λτ with respect to the norm inL∞([0, T ), L1((0, 1))). This immediately implies the result.
We nally obtain
Theorem 9. The limit measure λ of Lemma 6 and the limit function z satisfy∫ T
0
∫ 1
0
[z′′ · ζ ′′ + λ z′ · ζ ′ + (∂tz) · ζ
]ds dt = 0 ,
|z′| = 1 on [0, T )× [0, 1]
(WP2)
for all testfunctions ζ ∈ L2([0, T );H2((0, 1),R2)). This is the weak formulation of system
(10).
Proof. We consider a general variation ζ = δZnτ ∈ L2([0, T );H2((0, 1),R2)). Then, by
construction, the functions Znτ and λn
τ satisfy(δUn
τ + δU + δUL
)[Zn
τ ] ζ = 0 , (29)
with δUL[w]δw =∫ 10 λw
′ · δw′ ds for all ζ ∈ L2([0, T );H2((0, 1),R2))). As in the proofof Theorem 8, we may sum up (29) with the indices n = 0...bT/τc, write the resultingequation as an integral with respect to time where we have plugged in zτ and zτ andnally pass to the limit. As the limit equation we obtain (WP2). We might also split ζaccording to (22) and observe that the variation in admissible directions is zero by Theorem8, while the variation in non-admissible directions ψ disappears by the construction of λn
τ ,i.e. they satisfy (29). The sum of the two limit functions again is (WP2).
4.2 Nonlinear heat equations as gradient ows
cp [Ott01]
20
A Preliminaries from functional analysis (cp. [Eva98], [Eva90])
Let U ⊂ Rn,
Denition 6. Function spaces:
1. With the norm ‖u‖ := supU |u|, the set C(U,R) is the Banach space of bounded,continuous functions.
2. Let 0 < γ ≤ 1. A function u which satises |u(x) − u(y)| ≤ C|x − y|γ for allx, y ∈ U and a given C > 0 is said to be Hölder continuous. Its Hölder-seminorm
(Hölder-constant) is given by [u]C0,γ := supx,y∈U,x 6=y
|u(x)−u(y)||x−y|γ
.
3. The Hölder-space Ck,γ(U,R) is given by all functions, whose n-th derivatives are γ-Hölder-continuous. With the norm ‖u‖ :=
∑|α|≤n ‖Dαu‖C(U) +
∑|α|=n[Dαu]C0,γ(U)
it is a Banach space.
4. Suppose u, v ∈ L1loc and α a multiindex. We say that v is the α-th weak partial
derivative of u, written Dαu = v, provided∫U uD
αφdx = (−1)|α|∫U vφ dx for all
test functions φ ∈ C∞c (U). Ex: v(x) =
1 if 0 ≤ x ≤ 10 if 1 < x < 2
is the weak derivative of
u(x) =
x if 0 ≤ x ≤ 11 if 1 < x < 2
on U = (0, 2).
The Sobolev spaceW k,p(U) consists of all locally integrable functions u : U 7→ R suchthat for each multiindex α with |α| ≤ k, Dαu exists in the weak sense and belongs to
Lp(U). With the norm ‖u‖W k,p(U) =(∑
|α|≤k ‖Dαu‖pLp(U)
)1/pit is a Banach space.
5. For the Sobolev-spaces W k,2 we also write Hk(U). They are Hilbert-spaces. Notethat H0(U) = L2(U).
Denition 7. We say a sequence uk∞k=1 ⊂ X converges weakly to u ∈ X, written uk uif 〈u∗, uk〉 → 〈u∗, u〉 for each bounded functional u∗ ∈ X∗.
Properties
1. If uk → u, then uk u.
2. Any weakly convergent sequence is bounded.
3. If uk u, then ‖u‖ ≤ lim infk→∞ ‖uk‖
Theorem 10. (Weak compactness, Banach Alaoglu-theorem).
1. Banach-Alaoglu theorem: Bounded sets in a dual space of a Banach space (B∗ :=x∗ ∈ X∗ : ‖x∗‖ ≤ 1) are weak* precompact, i.e. ∃x∗∞ ∈ B∗ such that 〈x∗k, y〉 → x∗∞for all y ∈ B.This implies:
2. Let X be a reexive Banach space and suppose the sequence uk∞k=1 ⊂ X is bounded.
Then there exists a subsequence ukj∞j=1 and u ∈ X such that ukj
u. In other
words, bounded sequences in a reexive Banach space are weakly precompact.
21
3. Let 1 < p, q < ∞ and uk∞k=1 a bounded sequence in W k,p(U). Then there is a
subsequence and u∞ ∈W k,p(U) (reexive Banach space!) such that∫u
∑|α|≤n
(Dαukj−Dαu∞) Dαv dx→ 0
as j →∞ for all v ∈W k,q (the dual space of W k,p).
4. if (uk)∞k=1 is a sequence in L∞(U), then there exists a subsequence which converges
weakly* to a u∞ ∈ L∞, i.e.∫U (ukj
− u∞)v dx→ 0 as j →∞ for all v ∈ L1(U) (L∞
is the dual space of L1(U), but not vice versa!).
5. Assume that the sequence of signed measures µk∞k=1 is bounded in M(U), then
there exists a subsequence and a measure µ∞ ∈ M(U) such the µkj µ∞, i.e.∫
U φdµkj(x) →
∫U φdµ∞(x) for all φ ∈ Cc(U). ⇒ A bounded sequence of L1 func-
tions might concentrate and converge towards a δ-distribution in the sense of mea-
sures.
Theorem 11. (Morrey's inequality) Let n < p ≤ ∞. Let U ⊂ Rn such that ∂U is C1.
Let u ∈ W 1,p(U), then u has a version u∗ ∈ C0,γ(U) for γ = 1 − np with the estimate
‖u∗‖C0,γ(U) ≤ C‖u‖W 1,p(U) with a constant C, which only depends on p and n and U .
Denition 8. (compact embedding) Let X and Y be Banach spaces, X ⊂ Y . We saythat X is compactly embedded in Y , written X ⊂⊂ Y , provided
1. ‖x‖Y ≤ C‖x‖X for all x ∈ X and some constant C and
2. each bounded sequence in X is precompact in Y .
Theorem 12. Let U ⊂ Rn such that ∂U is C1 and let 1 ≤ p ≤ ∞ such that p 6= n.Then W 1,p(U) ⊂⊂ Lp(U). (If n < p ≤ ∞ this follows from Morrey's inequality and the
Arzela-Ascoli compactness criterion, otherwise from the Rellich-Kondrachov compactness
theorem).
References
[AGS05] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient ows in metric
spaces and in the space of probability measures, Lectures in Mathematics ETHZürich, Birkhäuser Verlag, Basel, 2005. MR MR2129498
[Amb04] Luigi Ambrosio, Steepest descent ows and applications to spaces of probability
measures, Lecture notes available on the author's website, July 2004.
[Eva90] Lawrence C. Evans, Weak convergence methods for nonlinear partial dierential
equations, CBMS Regional Conference Series in Mathematics, vol. 74, Publishedfor the Conference Board of the Mathematical Sciences, Washington, DC, 1990.MR MR1034481 (91a:35009)
[Eva98] , Partial dierential equations, Graduate Studies in Mathematics,vol. 19, American Mathematical Society, Providence, RI, 1998. MR MR1625845(99e:35001)
22
[GF65] I.M. Gelfand and S.V. Fomin, Calculus of variations., Revised English edition.Translated and edited by Richard A. Silverman. 3rd. printing. Englewood Clis,N.J.: Prentice-Hall, Inc. VII, 232 p. , 1965 (English).
[GH96] Mariano Giaquinta and Stefan Hildebrandt, Calculus of variations 1. The La-
grangian formalism., Grundlehren der Mathematischen Wissenschaften. 310.Berlin: Springer-Verlag. xxix, 475 p., 1996.
[H03] Lars Hörmander, The analysis of linear partial dierential operators i. distri-
bution theory and fourier analysis. reprint of the 2nd edition 1990., Classics inMathematics. Berlin: Springer. xiv, 440 p., 2003.
[LU68] Olga A. Ladyzhenskaya and Nina N. Ural′tseva, Linear and quasilinear elliptic
equations, Translated from the Russian by Scripta Technica, Inc. Translationeditor: Leon Ehrenpreis, Academic Press, New York, 1968. MR MR0244627 (39#5941)
[Ott01] Felix Otto, The geometry of dissipative evolution equations: The porous medium
equation., Commun. Partial Dier. Equations 26 (2001), no. 1-2, 101174 (En-glish).
[SS04] Takasi Senba and Takashi Suzuki, Applied analysis. Mathematical methods in
natural science., London: Imperial College Press. xiii, 378 p., 2004 (English).
[Str00] Michael Struwe, Variational methods, third ed., Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Re-sults in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveysin Mathematics], vol. 34, Springer-Verlag, Berlin, 2000, Applications to non-linear partial dierential equations and Hamiltonian systems. MR MR1736116(2000i:49001)
[Tes04] Gerald Teschl, Nonlinear functional analysis, Lecture notes available on the au-thor's website, 2004.
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