Vorlesung: Fortgeschrittene partielle Differentialgleichungen (PDG2) (SoSe 2015)



Content of the lecture (Kurzübersicht der Vorlesung):

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

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



14 April:

Introduction, practical Information (see main page).

Chapter 0: Recall PDG1 and motivation.

In PDG1 mainly studied: linear 2nd order eq's: elliptic, parabolic, hyperbolic. Main interest: Well-posedness (additionally: Properties of solutions).
However: Existence proved by writing down solutions for very specific equations and very limited geometries (with one exception: Perron's Method).
This course: Study methods to prove existence of solutions in more general set-up - still by doing for (more general) elliptic (parabolic/hyperbolic) linear 2nd order equations (but techniques often partly applicable in more general setting).
Two main ideas: (1) Extend the concept of 'solution' (beyond 'classical solution') and (2) Apply Functional Analysis (FA).

15 April:

Discussion of C^k-spaces and their deficiencies. Definition of Hölder-spaces.
Theorem: Hölder-spaces are Banach spaces (no proof).
Discussion: Hölder-spaces are good for many things in study of PDE, but because norm is defined via sup-norms, the estimates one needs to prove for the relevant operators often turn out to be hard to prove.
Instead: Use norms defined via integrals (L^p). Problem: C^k (or Hölder-spaces) with these norms will _not_ be Banach spaces.
Solution: 'Complete' the spaces. - Will not do this abstractly, but by introducing 'weak derivatives'. Hence, will prove existence of 'weak solutions'. Second part of method: Prove 'regularity' (ie that weak solutions are in fact classical solutions). Discussions.

Chapter 1: Weak derivatives and Sobolev spaces.

Motivation: Integrate against test-function and do partial integration.
Definition: Weak derivative.
[E] 253-256.

21 April:

Lemma: Uniqueness (as L^1_loc-function).
Examples (exercises; more later).
Definition: Sobolev spaces.
Remarks (to definition of Sobolev spaces).
Definition Sobolev norms. Remarks.
Definition: Convergence in Sobolev spaces.
Definition: W^{k,p}_0(U). Remarks.
[E] 257-259.

22 April:

Examples: Powers of |x|.
Theorem: Properties of weak derivatives.
Theorem: Sobolev spaces are Banach spaces.
Approximation of Sobolev functions by smooth functions: Discussion.
Theorem: Local approximation by smooth functions.
Theorem: Global approximation by smooth functions. (Proof next time.)
[E] 260-265.

28 April:

Proof of: Theorem: Global approximation by smooth functions.
Theorem: Global approximation by functions smooth up to the boundary. (End of proof next time.)
[E] 265-268.

29 April:

End of proof: Theorem: Global approximation by functions smooth up to the boundary.
Discussion/motivation: Extension and restriction of Sobolev functions.
A list of books either about Sobolev spaces, or containing a (more or less detailed) discussion of them, can be found here. See also the Lecture Notes from the course by Prof. Breit in WiSe 2013/14.

Theorem: Extension Theorem (extending W^{1,p}(U)-functions to W^{1,p}(R^n)-functions, with support in given V compactly containing U, and with control on the W^{1,p}-norm).

For higher order reflections (for W^{k,p}-extensions) see f.ex.
Haroske and Triebel Distributions, Sobolev spaces, elliptic equations, European Mathematical Society (EMS), Zürich (2008) p. 75-77.

[E] 268-271.

05 May:

Theorem: Trace Theorem (restricting ('taking the trace of') W^{1,p}-functions to the boundary gives an L^p-function, on the boundary, with control over the respective norms. When u is continuous, the restriction is the pointwise restriction. (All for bounded domains U with C^1 boundary.))
Theorem: For u in W^{1,p}(U), the trace of u on boundary of U is zero if and only if u is in W^{1,p}_c(U) (closure of C^infinity_c(U) in W^{1,p}). (Proof: Next time).
[E] 271-273.

06 May:

Proof of (easy part of): Theorem: For u in W^{1,p}(U), the trace of u on boundary of U is zero if and only if u is in W^{1,p}_c(U) (closure of C^infinity_c(U) in W^{1,p}).
Recap: Lecture this far.
Sobolev inequalites and Sobolev embeddings: Motivation and discussions.
Definition: For p in [1,n), the Sobolev conjugate to p (or the Sobolev exponent) is p^* = np/(n-p).
Theorem: Gagliardo-Nirenberg-Sobolev Inequality ('The Mother of all Sobolev inequalities'). (Proof next time.)
Remarks: Such an inequality can only hold for _one_ L^q, namley q=p^* (by scaling).
[E] 275-277.

12 May:

Proof of Gagliardo-Nirenberg-Sobolev inequality.
Theorem: Estimates for W^{1,p}(U) for p in [1,n) and U bounded (with boundary U in C^1).
Theorem: Estimates for W_0^{1,p}(U) for p in [1,p) and U bounded. Poincar'e inequality.
Remarks and dicussion.
[E] 277-280.

13 May:

Re-cap: Sobolev inequalities and embeddings for W^{1,p}, p < n.
Remark: p=n.
Theorem: Morrey's inequality. NO PROOF (but _statement_ is exam material - always the case, even if no proof given).
Definition: a 'version' of a function.
Theorem: Estimates for W^{1,p}(U) for p in (n,infinity] and U bounded (with boundary U in C^1). NO PROOF.
Remarks and discussions.
Theorem: General Sobolev inequalities: For W^{k,p}(U), U bounded (with boundary U in C^1). NO PROOF.
Remarks and discussions.
Definition: Continuous embedding of one Banach space in another. Compact embeddings.
Remarks and discussion.
Theorem: Rellich-Kondrachov (compact embedding in L^q for W^{1,p}, U bounded (with boundary U in C^1)). NO PROOF.
Remarks and discussion.
Re-cap: Sobolev spaces. Remarks on generalizations.

Chapter 2: Linear second order elliptic PDE.

Recall: Boundary value problem (will start by studying Dirichlet boundary condition).
[E] 280-289; 311.

19 May:

Definition: Linear second order PDO in _divergence_ form and in _non-divergence_ form.
Discussion and remarks.
Definition: _uniformly_ elliptic linear 2nd order PDO (in open set U).
Plan: (1) Prove existence of weak solutions (2) Study the regularity of weak solutions
(Aim: Prove weak solutions are classical solutions).
Discussions and remarks.
Weak solutions: Motivation (derivation of the weak formulation of boundary value problem).
Remarks and discussion.
Definition: The bi-linear form associated to a 2nd order elliptic operator in divergence form (with L^infinity coefficients), and of a _weak solution_ to the boundary value problem.
[E] 311-314.

20 May:

Theorem: Lax-Milgram (in real Hilbert space).
Remarks and discussion.
Theorem: Energy estimate (for bilinear form associated to uniform elliptic 2nd order PDO in divergence form).
Theorem: "First existence theorem for weak solutions": For 2nd order uniformly elliptic operator L in divergence form (with L^infinity - coefficients) there exists gamma such that for all mu > = gamma and all f in L^2(U) there exists unique weak solution u to Lu + mu u = f in U, u = 0 on dU.
[E] 315-319.

27 May:

Examples.
Definition: Formal adjoint L^* of 2nd order PDO L (with regularity condition on the 1st order term), and the adjoint bilinear form.
Remarks.

Overview of theory of compact operators:
Definition: Compact operator between Banach spaces X, Y.
Theorem: For compact operator K on a Hilbert space H, the adjoint K^* is compact.
Definition: Null-space N and range R of linear operator.
Definition: Resolvent set, spectrum, eigenvalue, eigenvector for A in B(X).
Remarks and dicsussions.
Theorem: Spectrum of compact operator (on infinitely dimensional Hilbert space H).
Remarks: Also for Banach space X.
Theorem: Eigenvectors of a compact _symmetric_ operator on a Hilbert space ('diagonalising').
Theorem: The Fredholm Alternative for compact operator on a Hilbert space H.
Remarks and discussions: Relation to finite dimensional case. Also for Banach space, but formulation more complicated.

Theorem: "Second Existence Theorem for weak solutions": The Fredholm Alternative. (End of proof next time.)
[E] 319-321.
For compact operators: [E] 721-722, 724-729.
[W] D. Werner, Funktionalanalysis, 7. Auflage, Springer (2011): 262-276.

02 June:

End of proof of "Second Existence Theorem for weak solutions": The Fredholm Alternative.
Further remarks and discussions: "Why" does the theorem hold?? Because the bilinear form B of the PDO L is bounded and (almost) coercive, and therefore (!!) the solution operator K is bounded from L^2 to H_^1_0. The compact embedding of H^1_0 in L^2 makes the solution operator _compact_ from L^2 in L^2. Then use (abstract) Fredholm Alternative (FA).

Theorem: "Third Existence Theorem for weak solutions": Existence of unique weak solution to
Lu = lambda u + f for _all_ f in L^2(U) _except_ if lambda is in the (real) spectrum Sigma of L (Sigma is a finite subset of R, or a real sequence going to + infinity).
Remarks and discussions: (1) "Why" does _this_ theorem hold? Because of the nature of the spectrum of the _compact_ solution operator.
(2) Eigenvalues and eigenvectors/-functions for Lu = lambda u with Dirichlet boundary condition. Helmholtz' equation.

Theorem: Boundedness of the inverse (for lambda not in Sigma). Proof next time.
Remarks: Two ways of re-writing inequality.
[E] 321-324.

03 June:

Remark: (Brief!) Discussion of complex equations and solutions.
Motivation: Discussion of inhomogeneous boundary value problem (BVP): How reduces to study of homogeneous BVP but with more general inhomogeneity in the equation.
Definition: The space H^{-1}(U).
Notation and remarks.
Definition: Norm on H^{-1}(U).
Theorem: Characterisation of H^{-1}(U). (Proof next time.)
[E] 324-325, 315, 299-300.

09 June:

Proof of Characterisation of H^{-1}(U) (from last time).
Definition: Weak solution to BVP with H^{-1}(U) - inhomogeneity.
Theorem: "Existence theorem for weak solutions": For 2nd order uniformly elliptic operator L in divergence form (with L^infinity - coefficients) there exists gamma such that for all mu > = gamma and all f in H^{-1}(U) there exists unique weak solution u to Lu + mu u = f in U, u = 0 on dU.
Remarks and discussion.
Next: Regularity of weak solutions.
Motivation and heuristics (more next time).
[E] 300, 314-315, 320, 326.

10 June:

Definition: Difference quotients of L^1_{loc} - functions.
Theorem: Difference quotients and weak derivatives.
Remarks and discussions.
Theorem: Interior H^2-regularity for weak (H^1) solutions to uniformly elliptic second order PDE's in divergence form.
Remarks and discussions.
Proof next time.
[E] 291-293,327-329.

16 June:

Proof of: Theorem: Interior H^2-regularity (see above).
[E] 327-331.

17 June:

Remarks: On (other) a priori estimates (Schauder and Calderon-Zygmund). "Continuity in data".
Theorem: Higher interior regularity for weak (H^1) solutions to uniformly elliptic second order PDE's in divergence form.
Theorem: Infinite differentiability in the interior (when coefficients and right side are infinitely differentiable).
Discussions and remarks: Result (and method) not optimal if not working in the C-infinity category - should use Schauder-theory.
Boundary regularity (ie "regularity up to the boundary of solutions to elliptic equations"): Discussion and motivation.
Theorem: Boundary H^2-regularity (proof next time).
Remarks and comments.
[E] 332-335.

23 June:

Proof of "Boundary H^2-regularity" (see above).
Theorem: Higher boundary regularity (proof next time).
Remark: "Continuity in data" if unique solution.
[E] 335-341.

24 June:

Proof of: Theorem: Higher boundary regularity (see above).
Theorem: Infinite differentiability up to the boundary.
Remark: Existence of classical solutions. That C-infinity not needed. Discussion of classical/weak solutions: no one-to-one correspondance: Example of a classical solution to a BVP which is _not_ a weak solution (since not in H^1) (Hadamard's example).
Remarks and discussions: Now can/could return to study of various properties of _classical_ solutions to _general_ uniformly elliptic equations (Maximum Principle, Harnack inequalities, regularity, a priori estimates, Mean Value Property, unique continuation etc etc - see PDG1) - properties which were studied for _harmonic_ functions in PDG1 (see summary in PDG1 after study of elliptic equations).
[E] 340-343; [A-U] 202-203.

30 June:

Example (of various properties of _classical_ solutions to _general_ uniformly elliptic equations): Maximum Principle (weak; strong).

Eigenvalues and eigenfunctions: Recall of definitions.
Special case we study: Symmetric operators (with b^i=c=0), with smooth coefficients.
Theorem: Existence of sequence of eigenvalues (going to infinity) and orthonormal basis (ONB) of eigenfunctions.
Remarks: Regularity of eigenfunctions, and classical solutions, "Dirichlet eigenvalues", Weyl's Law/Weyl-asymptotics, Spectral Geometry.
For Maximum Principles, see [E] 344-351; in particular Theorem 4 p. 350.
[E] 354-356.

01 July:

Definition: Principal eigenvalue of L (symmetric uniformly elliptic operator of second order).
Theorem: Variational principle for the principal eigenvalue. Rayleigh's Formula. The principle eigenvalue is non-degenerate/simple, and the (!) eigenfunction can be chosen to be strictly positive in U (and zero on the boundary) (end of proof next time).
Remarks: We talk about 'the positive ground state of L' (the eigenfunction). The quotient minimised in the variational description is called the Rayleight-Ritz quotient. The variational principle is _constrained_ minimisation.
[E] 356-360.

07 July:

End of proof: The principle eigenvalue is non-degenerate/simple.

Corollary: Existence of a sequence in H^1_0(U) which is orthonormal basis of L^2(U) (in the L^2-inner product) and orthogonal basis of H^1_0(U) (in the H^1-inner product).

Corollary: For a bounded set U, the optimal constant in the Poincare inequality is (lambda_1)^{-1/2}.
End of study of linear uniformly elliptic PDE's in divergence form. - Some outlook on possible generalisations and further questions (in the linear theory studied).

Chapter 3: Nonlinear elliptic equations.

(NOTE: This section has no proofs, and is NOT exam material).

Recall: "Linear", "semi-linear", "quasi-linear", and "Fully nonlinear" for 2nd order PDE's. The first three ones written in divergence form. Examples.
Programme: "Well-posedness" (existence, uniqueness, continuity in data).
Plan: (1) Define some suitable form of "generalised solution" and prove existence of such. (2) Study of regularity of "generalised" solutions.
Description of idea of: "Calculus of Variations", description of the 'Direct Method (in the Calculus of Variations') (more next time).
[E] 360; 2; 453-454.

Next semester: 'Semi-linear Elliptic PDEs (Lecture/Vorlesung, 2 SWS, ohne Übungen).

08 July:

First Variation and Euler-Lagrange (E-L) equation.
Lagrangian, functional. Heuristics: A minimiser satisfies the E-L eq (here, quasilinear 2nd order PDE in divergence form).
Examples: Dirichlet Principle (also seen in PDE1), Generalised Dirichlet Principle, Nonlinear Poisson equation, Minimal Surface eq.
Second Variation: Convexity condition (and relation to ellipticity).
Existence of minimisers: Discussion of coercivity, weak compactness, and weakly sequential lower semi-continuity.
Theorem: Existence of minimisers with coercivity & convexity condition on the Lagrangian.
[E] 453-459, 465-471.

13 July:

Theorem: If strict convexity, uniqueness of solution.
Theorem: Under certain growth conditions on the Lagrangian and derivatives: A minimiser is a weak solution to the E-L eq.
Theorem: Under strong enough convexity conditions: A weak solution to the E-L eq is necessarily a minimiser.
Constrained minimisation and Lagrange multipliers: Recall characterisation of the Principal Eigenvalue (for a symmetric 2nd order operator on a bounded domain) - is example of constrained minimisation.
Theorem: Existence of minimiser under constrained minimisation, under growth condition on the non-linearity in the constraint.
Theorem: A minimiser satisfies E-L eq with Lagrange multiplier.

Critical Points, Nonlinear Analysis and Nonlinear FA: Brief discussion.
Mountan Pass Theorem: Statement & heuristics.
[E] 471-475, 488-491, 505-506.

14 July:

Non-variational Methods: Monotonicity Methods - motivation and heuristics.
Definition: Monotone vector field a.
Theorem: For a monotone, coercive, and maximally linearly growing vector field a: The Dirichlet BVP for the nonlinear eq -div a(DU) = f has a weak solution.
Theorem: If the vector field is strictly monotone, then the weak solution is unique.
Description of proof of existence: Via Galerkin approximation: Solve approximate problem on finite dimensional linear subspace, and pass to the limit (via a priori estimates). Existence for finite dim problem: Use Brouwers' Fixed Point Theorem (nonlinear analysis).
Regularity: Discussion of H^2-regularity for solution to (simple) E-L eq's ('Second derivatives for minimisers').
Discussion of higher regularity: More complicated in nonlinear case than in linear case; (brief!) discussion of De Giorgi/Nash/Moser theory and Schauder estimates.
[E] 527-533, 463, 482-487.

End of Lectures!


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

Letzte Änderung: 14 July 2015 (no more updates).

Thomas Østergaard Sørensen












Home
Teaching
Publications
Curriculum Vitae