Partielle Differentialgleichungen (WiSe2013/14)

Vorlesung: Partielle Differentialgleichungen (PDG1) (WiSe 2013/14)

Kurzübersicht der Vorlesung:

[E] L. C. Evans, Partial Differential Equations: Second Edition, AMS, Providence, RI, 2010.

[A-U] W. Arendt und K. Urban, Partielle Differenzialgleichungen, Spektrum Akademischer Verlag, 2010.

14. Oktober:

Introduktion, praktische Informationen.

Kapitel 0: Einleitung, Motivation, Ziele.

Wo PDGs auftauchen (Physik, Chemie, Ingeniörwissenschaften, Medizin etc etc). Keine PDGs im Kurs herleiten. Kurze Diskussion von Modellierung (durch PDGs). Notation: partielle Ableitungen, Multiindices. Definition von eine PDG.
[A-U] 1-10, 17-28. [E] xvii-xxi; 1-2.

17. Oktober:

Diskussion vom Lösen von PDGs und der Vielfalt an PDGs.
Aufteilen von PDGs in: lineare, semi-lineare, quasi-lineare und vollkommen nicht-lineare PDGs. Diskussion.
Definition von System von PDGs.
Erste Beispiele (lineaere/nicht-lineaere, Gleichungen/Systeme):
Laplace-Gleichung, Helmholtz-Gl, Lineare Transport-Gl, Liouville's Gl, Wärmeleitungs-Gl, Schrödingers Gl, Wellengleichung, Eikonal-Gl, Nichtlineare Poisson-Gl, Minimalflächen-Gl, Maxwell-Gl, Navier-Stokes-Gl.
[A-U] 28-30, 51. [E] 2-6.

21. Oktober:

'System des Tages': Euler-Gleichungen.
Strategien/Ziele beim Studium von PDGs: Explizit Lösen! Wenn nicht möglich (am meisten der Fall), die Korrektgestelltheit (im Sinne von Hadamard) studieren (definiert und diskutiert).
Definition und diskussion von klassische Lösungen.
Diskussion von Klassifikation in linear/nichtlinear, Gleichung/System, nach Ordnung, und deren Schwierigkeitsgrad.
Klassifizierung von linearen PDGs 2. Ordnung mit konstanten (und nicht-konstanten) Koeffizienten im Fall von 2 Variablen: Elliptisch, parabolisch, hyperbolisch. Diskussion.
[A-U] 28-30, 34, 43-45. [E] 6-9.

24. Oktober:

'Gleichung des Tages': Fokker-Planck-Gleichung.
Definition von elliptisch, parabolisch, hyperbolisch im Fall von linearen PDGs 2. Ordnung mit nicht-konstanten Koeffizienten (im allgemeinen Fall von n Variablen). Wichtigste Beispiele: Poisson-, Wärmeleitungs- und Wellengleichung.

Kapitel 1: Transportgleichungen.

Ziel: Illustration von Charakteristikenmethode, und von Anfangs- und Randwerteproblemen.
Beispiel: Transportgleichung mit konstanten Koeffizienten auf R^n x (0,infinity). Lösung durch Charakteristikenmethode (Lösung muss konstant auf gewisse Geraden sein).
Beispiel: Leitet zur Anfangswertproblem. Lösung davon durch Charakteristikenmethode.
Moral der Geschichte, was man daraus lernen soll, und Diskussion.
Beispiel: Transportgleichung mit konstanten Koeffizienten auf (0,1) x (0,T).
[A-U] 45-47, 34-37. [E] 18.

28. Oktober:

The language of the course was changed to English. Information (on the webpages) published in German this far will not be translated.

'Equation of the Day': Monge-Ampère Equation.
Linear transport equation (n=1) on a finite interval, in x and in t. Diskussion of Method of Characteristics, and of prescribable boundary values. Same for halfline in x and halfline in t.
Solution by the Method of Characteristics of the n-dimensional linear inhomogeneous transport equation with constant coefficients in half space.

Chapter 2: The Laplace Equation.

The Laplace and Poisson equations. Definition of harmonic functions (in open set U). Fundamental solutions for the Laplace equation in n >= 2. Diskussion, remarks.
Theorem: Solution of Poisson's Equation in R^n with inhomogeneity f in C^2_c (proof next time).
[A-U] 37. [E] 19, 20-23.

31. Oktober:

'Equation of the Day': Euler-Tricomi Equation.
Proof from last time. Discussion.
Theorem: Mean-Value Properties (MVP) of harmonic functions.
Theorem: Converse to Mean-value properties.
[E] 23-26.

For surface integration, Gauss's theorem, Green's identities etc, see for example:
[A-U] 220-230,
H. Amann and J. Escher, Analysis III, Birkhäuser Verlag (2001),
W. Walter, Analysis 2, 5. erweiterte Aufl., Springer Verlag (2002) (in particular, Aufgabe 9 p. 307).

04. November:

'Equation of the Day': p-Laplacian Equation.
Strong Maximum (and Minimum) Principle for the Laplace equation. Discussion.
Definition of the Dirichlet Problem (Boundary Value Problem (BVP)) for the Laplace and the Poisson equation.
Consequence of the max. principle: Strict positivity of solutions of the Dirichlet Pb for the Laplace equation (on _bounded_ sets) with positive (strict positive _somewhere_) boundary data.
Uniqueness of (classical solutions to) the Dirichlet Pb for the Poisson equation on _bounded_ sets.
Continuous dependence of the solutions to the Dirichlet Pb for the Laplace equation (on _bounded_ sets) on the boundary data.
Regularity Theorem: Any C-2-function with the Mean Value Property is C-infinity (ie smooth). Hence any harmonic function is C-infinity.
[A-U] 71-73. [E] 26-28.

07. November:

'Equations of the Day (system): Linear (stationary) elasticity. equations.
Proof of Regularity Theorem (from last time). Discusssion.
A priori (pointwise) estimate on all derivatives of a harmonic function. Discussion.
Consequence: Liouville's Theorem: Only functions bounded and harmonic on R^n are constants. Discussion: Compare Liouville's Theorem (and proof!) in Complex Analysis.
Consequence: For n>=3: Any _bounded_ solution of Poisson's Equation in R^n with inhomogeneity f in C^2_c is the one given earlier (f convolved with the Fundamental Solution) _plus_ a constant.
Regularity Theorem: Any harmonic function is (real) analytic (finish proof next time).
[E] 28-31.

11. November:

'Equations of the Day (system): Evolution equations of linear elasticity.
Proof of analyticity of harmonic functions.
Theorem: Harnack's Inequality for harmonic functions.
Discussion of solving the Dirichlet Problem for the Poisson equation.
Definition: Green's function for a bounded, open subset of R^n. Discussion.
Theorem: _If_ u is C^2 in closure of U, and _if_ G is Green's function for U, and _if_ u solves the Dirichlet Problem for the Poisson equation (with f in C(U), g in (C(dU)), _then_ u is given by an integral formula involving g integrated against dG on dU, and f integrated against G on U.
Discussion: Leaves _one_ candidate as solution _under_ certain conditions.
[E] 31-35.

14. November:

'Equation of the Day': Kolmogorov's Equation.
Proof of Theorem stated last time (see above). Discussions.
Theorem: Green's function is symmetric: G(x,y) = G(y,x) for all x,y in U, x noteq y.
[E] 33-36.

18. November:

'Equation of the Day': Scalar reaction-diffusion equation.
Derivation of Green's function for half-space by the method of mirror-images. Poisson's Kernel, and Poisson's Formula. Discussion.
Theorem: For g continuous and bounded on R^{n-1}, Poisson's Formula gives a solution to the Dirichlet problem for Laplace's equation on half-space, with g as boundary value.
Derivation of Green's function for the unit ball.
[E] 37-39.

21. November:

'Equation of the Day': Klein-Gordon equation.
Definition of Green's function for unit ball B(0,1).
Definition of Poisson's kernel and Poisson's Formula for ball B(0,r).
Theorem: For g continuous and bounded on dB(0,r) (sphere), Poisson's Formula gives a solution to the Dirichlet problem for Laplace's equation on the ball B(0,r), with g as boundary value. (Proof: Exercise sheet.)
Discussion: Have proved well-posedness of Dirichlet problem for Laplace's equation on ball B(0,r).
Weyl's Lemma: If u is continuous on U and a _weak_ solution to Delta u=0, then u is harmonic in U. (Motivation and proof).
[E] 39-41.
[H-L] Q. Han und F. Lin, Elliptic Partial Differential Equations: Second Edition, AMS (2011), S. 6-8.

25. November:

'Equation of the Day': General wave equation.
Aim: Perron's Method (to solve the Dirichlet problem for the Laplace equation on any bounded U) - relies on Maximum Principle & solvability of the Dirichlet problem for the Laplace equation on any _ball_.
Definition of sub-harmonic functions via Delta v (for v in C^2) (see Problem Sheet 4).
Lemma: Alternative characterization of sub-harmonic functions.
Defintion: Sub-harmonicity (and super-harmonicity) for v only in C(U), via the alternative chracterization in lemma.
Lemma: (Strong) Maximum Principle for u subharmonic, v super-harmonic, when set U is open, connected, and bounded.
Presentation/motivation of/for Perron's Method.
[H-L] 125-126; [H] Q. Han, A Basic Course in Partial Differential Equations, AMS (2011), S. 126-128.

28. November:

'Equation of the Day': sine-Gordon equation.
Discussion of Perron's Method.
Lemma: Harmonic lifting in ball B of subharmonic function v.
Lemma: A sequence of harmonic functions in U, which is uniformly bounded on compact subsets of U, has a subsequence which converges uniformly on compact subsets of U, to a harmonic function. (Proof next time.)
Proposition: Given U open and bounded, g continuous on dU (boundary of U), then the supremum (called u_g) over all v, subharmonic in U and bounded by g on dU, is harmonic in U.
[H-L] 126-128; [H] 128-131; 104.

02. December:

'Equation of the Day': Scalar Conservation Law.
Proof of Lemma from last time (see above).
Proposition: Given U open and bounded, g continuous on dU (boundary of U), x^0 on dB, then the supremum (called u_g(x)) over all v, subharmonic in U and bounded by g on dU, converges to g(x^0) for x -> x^0 (x in U) IF there exists a barrier function at x^0. Discussion.
Definition: Barrier function at x^0, x^0 Dirichlet-regular, dU (boundary U) Dirichlet-regular.
Theorem: U open, bounded, with dU Dirichlet-regular. Then the Dirichlet-problem for the Laplace equation has a unique solution for all g continuous on dU.
Corollary: U open, bounded, with dB Dirichlet-regular. Then Green's function for U exists. Discussion.
Definition: Exterior ball condition at x^0 in dU.
Proposition: U open, x^0 in dB, U satisfying ext. ball condition at x^0. Then x^0 is Dirichlet-regular (for U). (Proof next time.)
Corollary: U open, bounded, U satisfying ext. ball condition for all x in dU. Then the Dirichlet-problem for the Laplace equation has a unique solution for all g continuous on dU.
[H] 104; 131-133; [H-L] 128-130.

05. December:

'Equation of the Day': System of conservation laws.
Proof of Proposition from last time (see above).
Discussion and remarks: More on when set U is Dirichlet-regular in Exercises: If dU (boundary) is C^2 then Dirichlet-regular. If U convex, then Dirichlet-regular.
Energy methods: Techniques involving L^2-norms.
Theorem: Uniqueness of solution to Dirichlet boundary value problem (BVP) for the Poisson equation (via Energy Method. Done earlier by Max. Principle).
Dirichlet's Principle: Motivation.
Theorem: u solves the BVP for the Poisson equation if and only if u minimises an energy functional over a certain admissible set.
Remarks on problems of this approach: The admissible set not a good set to study the minimisation problem.
Resume of topics studied for _elliptic_ equations (more next time).
[H] 132; [E] 41-43 (see also Chapter 8).

09. December:

'Equation of the Day': Porous medium equation.
Resume of topics studied for _elliptic_ equations (continued from last time). Discussion and remarks.

Chapter 3: The Heat Equation.

The heat equation (also called diffusion equation) and the inhomogeneous heat equation; notation and definitions.
'Principle': Statements on harmonic functions often have similar, but more complicated version about solutions to the heat equation.
Fundamential solution for the heat equation. Remarks on properties.
Lemma: Integral over x of fundamental solution equals 1 for all t>0.
Discussion of initial value problem/Cauchy problem (compare Chapter 1).
Theorem: Solution of the Cauchy problem for the heat equation is given by convolution of the fundamental solution with the initial value g. (End of proof next time.)
[E] 44-47.

12. December:

'Equation of the Day': Airy's equation.
End of proof from last time (see above).
Infinite propagation speed (for the heat equation): If a non-negative initial temperature distribution g (continuous and bounded) is strictly positive somewhere, then the temperature u (solution to the heat equation) is strictly positive everywhere in space for any later time t (no matter how small). Discussion (shall see contrast to wave equation later).
Duhamel's Principle: To solve _inhomogeneous_ heat equation (with zero initial value) with right side f, solve _homogeneous_ initial value problem (starting at s>0) with f( ,s) as initial data (at t=s), then integrate solutions over s.
Theorem: Solution of the inhomogeneous heat equation with zero initial value is given by Duhamel's principle: Convolve f with the fundamental solution, in all of R^n, and from 0 to t.
[E] 47-51.

16. December:

'Equation of the Day': Korteweg-de Vries equation.
Corollary: Solution of inhomogeneous heat equation with initial value g is sum of the solutions from Theorem 3.3 and Theorem 3.5.
Definition: Parabolic cylinder U_T and parabolic boundary Gamma_T (for U open and bounded in R^n and T>0).
Mean-value property: Definition of 'heat ball' (whose boundary is level set for the fundamental solution of the heat equation).
Theorem: Mean-value property for solution to the heat equation.
Theorem: Strong maximum principle for solution to the heat equation in U_T (U bounded). (End of proof next time.)
[E] 51-56.

18. December:

'System of the Day': Reaction-diffusion system.
End of proof from last time (see above).
Remark: Consequence of Max. Principle: Infinite propagation speed for solution to the heat equation in U_T (U bounded) with non-negative (and _somewhere_ positive) initial temperature distribution g (continuous).
Another consequence:
Theorem: Uniqueness of (classical) solutions to the _inhomogeneous_ heat equation (with continuous data f,g) in U_T (U bounded).
Theorem: Maximum Principle for Cauchy Problem (IVP in U=R^n) for the heat equation _with_ growth condition (u bounded by A e^{a|x|^2}).
Consequence:
Theorem: Uniqueness of (classical) solutions to the _inhomogeneous_ heat equation (with continuous data f,g) in R^n x [0,T] _with_ growth condition (|u| bounded by A e^{a|x|^2}).
Discussion: Existence of infinitely many _non_physical (very rapidly growing) solutions to the Cauchy Problem with zero initial value.
Theorem: If u in C^2_1(U_T) solves heat equation in U_T, then u is smooth (C-infinity). (Proof next time).
[E] 56-59.

19. December:

'Equation of the Day': Inviscid Burgers' equation.
Proof of smoothness of solutions to the heat equation (for t>0) in the case u in _C^3_ (from last time; see above).
Comments and remarks.
Theorem: A priori estimates (in heat cylinders) on derivatives of solutions to the heat equation.
Remarks: On analyticity.
[E] 59-62.

(Compare also [H] 158-197.)

Merry Christmas & Happy Newyear!

09. January:

'Equation of the Day': Telegraph equation.
Energy methods:
Theorem: Uniqueness of solutions to inhomogeneous heat equation with given initial and boundary value (U bounded).
Theorem: Backwards uniqueness (in time) for solutions to heat equation with prescribed boundary values (U bounded) (but _not_ prescribed initial value).
Resume of topics studied for _parabolic_ equations. Discussion and remarks.

Chapter 4: The Wave Equation.

[E] 62-65.

13. January:

'Equation of the Day': (Viscous) Burgers' equation.
The wave equation and the inhomogeneous wave equation; notation and definitions. Remarks and discussion of wave phenomenon. Discussion of differences to Laplace and Heat equation.
Solution of wave equation for n=1 (in R x (0,infinity) ) via d'Alembert's Formula. (Derived by solving two 1-dim transport equations, one homogeneous, and one inhomogeneous). (For another derivation, see exercises.) Discussion of right-moving and left-moving waves.
Solution of wave-equation for n=1 on the half-line (in (0,infinity) x (0,infinity) ) via Reflection Method (odd reflection the the origin).
Outline: Goal: Solve wave equation (in R^n) for n>=2.
Plan: First for n=3,2 (later for all odd, even dim's). For n=2, use result for n=3 by leaving out variable. For n=3, use Method of Spherical Means, to derive PDE for these means, then transform this PDE into wave equation for n=1 on the half-line (in (0,infinity) x (0,infinity) ) (which just solved).
[E] 65-70.

16. January:

'Equation of the Day': Beam equation.
Lemma: In any dimension n, the spherical mean of a solution of the wave equation in R^n solves the Euler-Poisson-Darboux equation (in (r,t) in (0,infinity) x (0,infinity)) (including: initial values are spherical means of initial values of wave equation).
Lemma: If U solves Euler-Poisson-Darboux equation for n=3 in (0,infinity) x (0,infinity) with initial values G,H, then tilde(U) solves wave equation for n=1 on the half-line (in (0,infinity) x (0,infinity), with initial values tilde(G), tilde(H), where tilde(f)(r) = r f(r).
Theorem: Kirchoff's Formula for solution of wave equation for n=3 (involving spherical means of the initial data g,h).
"Method of descent": Solving wave equation for n=2 via 'descending' the formula for n=3 (continuation next time).
[E] 70-73.

20. January:

'Equation of the Day': Bi-harmonic equation.
Continued: Solving wave equation for n=2 via 'descending' the formula for n=3.
Theorem: If u solves the wave equation for n=2 then it satisfies Poisson's Formula.
Solution of the wave equation in R^n for n odd (>= 3). (Continuation next time).
[E] 73-76.

23. January:

'Equation of the Day': Hamilton-Jacobi equation.
Theorem: Formula for solution of wave equation in R^n for n odd (>= 3) (involving spherical means of the initial data g,h).
Discussion and remarks: Finite propagation speed.
"Method of descent": Solving wave equation in R^n for n even (>=2) via 'descending' the formula in R^k for k=n+1 odd.
Theorem: Formula for solution of wave equation in R^n for n even (>= 2) (involving spherical means of the initial data g,h).
[E] 77-80.

27. January:

'Equation of the Day': Nonlinear wave equation.
Remarks: Huygens' Principle, Range of influence, and Domain of dependence, and differences in odd (n >=3) and even dimension. Finite propagation speed.
IVP for nonhomogeneous wave equation; Duhamel's Principle:
Theorem: The solution of the IVP (zero IV's) for the nonhomogeneous wave equation is given by Duhamel's Principle.
Energy Methods:
Theorem: Uniqueness of C^2 - solutions to the initial/boundary value problem on U_T (for bounded, open domain U in R^n and T>0).
Theorem: Finite propagation speed/domain of dependence (by energy method; proof next time).
[E] 80-84.

30. January:

'Equation of the Day': Nonlinear Schrödinger equation.
Proof of Theorem on finite propagation speed/domain of dependence from last time.
Corollary: Uniqueness C^2 - solutions to the initial value problem for the inhomogeneous wave equation in R^n.
Discussion and remarks.
Resume of topics studied for _hyperbolic_ equations. Discussion and remarks.

Discussion: 2nd order linear equations - and relation to (most) 'Equations of the Day'.
(Most of the rest: related to _transport_ equation.)
Discussion: Need _much_ more on _existence_ (ie, _methods_ to prove existence). Needs extending the concept of 'solution' to various types of 'generalised solutions' (weak, viscosity, entropy, distributional etc). For more, see PDE 2 in SoSe 2014. (Also, see seminar on 'Pseudodifferential Operators' in SoSe 2014 - old description here.)
[E] 83-84.

03. February:

'System of the Day': Navier-Stokes equation.

Chapter 5: Method of Characteristics.

Derivation of _characteristic equations_ (system of ODE's) for general (nonlinear) first order PDE:
F(Du,u,x) = 0 in U (in R^n).
Definition of 'characteristics' and 'projected characteristics'.
Example: Linear homogeneous transport equation (non-constant coefficients).
Example: Quasi-linear (first order) PDE.
Straightening the boundary of U (locally).
[E] 96-103; [H] 16-30.

06. February:

'System of the Day': Einstein's Field Equations (from General Relativity.)
Compatibility conditions on boundary data (for solving characteristic equations).
Non-characteristic boundary data. Ensures admisibility locally near admissible data.
Local solution: Solving characteristic equations in vicinity of admissible data, inverting these with respect to initial point (on boundary) and time, then define u via transporting boundary data via projected characteristics, _gives_ (locally, near _part_ of the boundary) solution to the boundary value problem.
[E] 103-106; [H] 16-30. (See also [E] 106-108 for proof of last theorem, and [E] 109-114 for examples and applications.)

End of Lectures!

-----------------------------------

Letzte Änderung: 18. February 2014 (no more updates).

Thomas Østergaard Sørensen

Teaching