Partial Di erential Equations - KIT - Fakultät für … 2 24.10.2013 Tobias Lamm - PDE Institute for Analysis jKIT KIT 1) Harmonic functions: Mean value property, fundamental solutions,

0 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT

KIT

Institute for Analysis |KIT

Partial Differential Equations

Tobias Lamm

KIT – University of the State of Baden-Wuerttemberg andNational Research Center of the Helmholtz Association www.kit.edu

Overview


KIT

Content

Literature

What is a PDE?

Examples

Content


KIT

1) Harmonic functions: Mean value property, fundamental solutions, maximumprinciple, energy estimates, Perron method

2) Heat equation: initial value problem, maximum principle, regularity, Harnackestimate

3) Wave equation: solution formulas in 1, 2, and 3 dimensions

4) Maximum principles for general elliptic operators

5) Sobolev spaces and L2 theory: Existence and regularity of solutions ofelliptic PDE’s

Content


KIT






Content


KIT






Content


KIT






Content


KIT






Overview


KIT

Content

Literature

What is a PDE?

Examples

Literature


KIT

L.C. Evans: Partial Differential Equations, 2nd edition (Chapters 1,2,3,5)

D. Gilbarg and N. Trudinger: Elliptic Partial Differential Equations of secondorder (Chapters 4, 5)

Q. Han and F. Lin: Elliptic Partial Differential Equations (Chapters 1, 4)

R. Courant and D. Hilbert: Methods of Mathematical Physics (Motivation andExamples)

Literature


KIT





Literature


KIT





Literature


KIT





Overview


KIT

Content

Literature

What is a PDE?

Examples

What is a PDE?


KIT

Definition: (see Evans)Let u ∈ Ck (Ω) be a k-times continuously differentiable function on a domain

Ω ⊂ Rn and let F : Rnk ×Rnk−1 × · · · ×Rn ×R×Ω→ R be given. Anexpression of the form

F (Dku(x), Dk−1u(x), . . . , Du(x), u(x), x) = 0, x ∈ Ω (1)

is called a Partial Differential Equation of order k .

Types of PDE’s


KIT

A PDE is called linear if

∑|α|≤k

aα(x)Dαu = f (x), α = (α1, α2, . . . , αn)

for given functions aα and f .

A PDE is called semilinear if the equation is linear in the highestderivatives, i.e.

∑|α|=k

aα(x)Dαu + a0(D

k−1u, Dk−2u, . . . , Du, u, x) = 0

A PDE is called quasilinear if

∑|α|=k

aα(Dk−1u, . . . , Du, u, x)Dαu + a0(D

k−1u, . . . , Du, u, x) = 0.

Types of PDE’s


KIT


∑|α|≤k

aα(x)Dαu = f (x), α = (α1, α2, . . . , αn)



∑|α|=k

aα(x)Dαu + a0(D

k−1u, Dk−2u, . . . , Du, u, x) = 0


∑|α|=k

aα(Dk−1u, . . . , Du, u, x)Dαu + a0(D

k−1u, . . . , Du, u, x) = 0.

Types of PDE’s


KIT


∑|α|≤k

aα(x)Dαu = f (x), α = (α1, α2, . . . , αn)



∑|α|=k

aα(x)Dαu + a0(D

k−1u, Dk−2u, . . . , Du, u, x) = 0


∑|α|=k

aα(Dk−1u, . . . , Du, u, x)Dαu + a0(D

k−1u, . . . , Du, u, x) = 0.

Overview


KIT

Content

Literature

What is a PDE?

Examples

Examples I


KIT

Poisson equation, u, f : Ω→ R

∆u :=n

∑i=1

∂2iiu = f

Transport equation

ut + b ·Du = ut + bi∂iu = 0, b ∈ Rn

Eigenvalue equation (λ ∈ R)

∆u = λu ∈ Ω

Examples I


KIT


∆u :=n

∑i=1

∂2iiu = f

Transport equation



∆u = λu ∈ Ω

Examples I


KIT


∆u :=n

∑i=1

∂2iiu = f

Transport equation



∆u = λu ∈ Ω

Examples II


KIT

Heat equation, u : Ω× (0, T ]→ R, T > 0

ut − ∆u = 0

Schrodinger equation u : Ω× (0, T ]→ C, T > 0

iut + ∆u = 0

Wave equation, u : Ω× (0, T ]→ R, T > 0

utt − ∆u = 0

Examples II


KIT


ut − ∆u = 0


iut + ∆u = 0


utt − ∆u = 0

Examples II


KIT


ut − ∆u = 0


iut + ∆u = 0


utt − ∆u = 0

Examples III


KIT

Reaction-Diffusion equation, harmonic map equation into sphere

ut − ∆u = f (u), −∆u = u|Du|2 semilinear

Graphical minimal surface equation

div

Du√1 + |Du|2

= 0 quasilinear

Monge-Ampere equation

det(D2u) = f fully nonlinear

Examples III


KIT




div

Du√1 + |Du|2

= 0 quasilinear



Examples III


KIT




div

Du√1 + |Du|2

= 0 quasilinear



Documents

Partial Di erential Equations - KIT - Fakultät für … 2 24.10.2013 Tobias Lamm - PDE Institute for Analysis jKIT KIT 1) Harmonic functions: Mean value property, fundamental solutions,