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 |
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