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
1 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
KIT
Content
Literature
What is a PDE?
Examples
Content
2 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
2 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
2 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
2 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
2 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
Overview
3 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
KIT
Content
Literature
What is a PDE?
Examples
Literature
4 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
4 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
4 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
4 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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)
Overview
5 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
KIT
Content
Literature
What is a PDE?
Examples
What is a PDE?
6 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
7 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
7 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
7 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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.
Overview
8 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
KIT
Content
Literature
What is a PDE?
Examples
Examples I
9 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
9 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
9 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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 II
10 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
10 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
10 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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 III
11 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
11 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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
11 24.10.2013 Tobias Lamm - PDE Institute for Analysis |KIT
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