Vorlesung: Viscosity Solutions for nonlinear PDEs (WS 2014/15)Content of the lecture: 07 October: Introduction, practical Information (see main page). Chapter 0: Introduction and motivation. Notation. General PDE of order k; general PDE of order _2_. Definition: 'classical solution'. Special cases: quasi-linear, semi-linear, linear. Classification (of linear eq's) in elliptic, hyperbolic, parabolic. Examples: Laplace, Poisson, wave, and heat eq's. In this course: Study 'Fully nonlinear elliptic 2nd order PDEs' ('elliptic' to be defined later. - Also, _some_ on parabolic). Linear equations in _divergence_ and _non-divergence_ form. Discussion: "weak solutions": For _divergence_ form (mostly). 'Viscosity solutions': For non-divergence - but especially for _really_ nonlinear_. Examples: Linear (non-divergence form), Hamilton-Jacobi (H-J) eq (example: eikonal eq), Bellman eq, Bellman-Isaacs eq, level-set mean curvature flow eq, Monge-Ampere eq, Parabolic eq, infinity-Laplace eq, k-Hessian eq. A 'problem' in PDE consists of: Equation PLUS extra condition. - We shall study Dirichlet Boundary Value Problem (BVP): Value on boundary of domain is given. 14 October: Discussion/motivation: (Hadamard) Well-posedness: Existence, uniqueness and stability/continuity in data. Difficulty: Existence of _classical_ solutions. Approach: Define 'useful' concept of 'generalised solution' (in PDE2/for 2nd order linear elliptic eq's in div form: weak derivatives, Sobolve spaces, weak solutions) - here: 'viscosity solutions'. Then prove existence of generalised solutions, and that generalised solutions which are smooth enough are in fact classical solutions. Reduces study to: existence, uniqueness, stability, regularity of _viscosity_ solutions. Overview of course. Chapter 1: Definition and properties of viscosity solutions. Always: Omega open and bounded domain of R^n. Always: F continuous. Definition: _classical_ subsolution, supersolution, solution of fully non-linear PDE. Comparison with subharmonic, superharmonic, harmonic. Definition: F is (degenerate) elliptic iff it is non-increasing in (the matrix) X. Remarks and discussions. Definition: F is proper iff it is elliptic and non-decreasing in r. Remarks and discussions. Exercise! (DO!) Explanation of the name '(vanishing) viscosity solution' (to be continued next time). 21 October: End of explanation of the name '(vanishing) viscosity solution' on 'tautological example'. Proposition: For fully nonlinear elliptic 2nd order equation, add a 'vanishing viscosity'- term. If the subsolutions to this equation converge uniformly on compact sets, then the limit function u satisfies: For all phi in C^2, with max of u - phi at x, we have F(x,u(x),D phi(x), D^2 phi(x)) <= 0. This gives 'proper' explanation for the name '(vanishing) viscosity solution', and motivates the later _definition_ of 'viscosity solution'. 28 October: Another motivation for the (later!) _definition_ of 'viscosity solution': Using the Maximum Principle on classical sub/super/solition of -Delta u = 0 (or more generally of -tr(A D^2 u) = 0, A >= 0). Definition of viscosity subsolution/supersolution/solution. Remarks: On assumptions. Proposition: Equivalent formulation of the definition of viscosity sub/super/solution. Proposition: Assume F is elliptic. A function u is a classical sub/super/solution iff it is a viscosity sub/super/solution AND is C^2. Remark: Reduces existence question for classical solutions to (1) existence of viscosity solutions (2) _regularity_ of viscosity solutions. Definition: Upper and lower semi-continuous functions, upper and lower semi-continuous envelopes of functions. Properties. Notation. Remarks. 04 November: Proposition: If u is a viscosity subsolution (resp. supersolution) on Omega, bounded from above (resp. from below) on Omega, then it is a viscosity subsolution (resp supersolution) on any open subset Omega' of Omega. Chapter 2: Comparison Principles. Formulation and discussion of 'Comparison Principle'. Proposition: Comparison Principle implies uniqueness of solutions to Dirichlet BVP. Equations we will prove Comparison Principles for are all on form nu u + F(x,Du,D^2 u) = 0. Example: _Will_ need conditions on F to prove Comparison Principle: A first order Dirichlet BVP with _two_ (classical!) solutions. Definition: Pucci operators (aka Pucci's Extremal Operators). Proposition: Properties of Pucci operators. Definition: Uniform ellipticity of F = F(x,p,X). Remarks; discussion of 2nd order linear case: _IS_ 'uniform ellipticity'. Overview of results in Chapter 2 (ie of assumptions on F for which we can/will prove Comparison Principle): (1) 'Classical comparison' (one classical, one viscosity sub/super-solution):
Proposition 1: 'Classical comparison' (one classical, one viscosity sub/super-solution): Any elliptic F and nu > 0. Remarks. Proposition 2: 'Classical comparison' (one classical, one viscosity sub/super-solution): Uniformly elliptic F (+ ellipticity condition in Du) and nu >=0 (nu > 0 covered by Proposition 1 via Lemma: F uniformly elliptic implies F (degenerate) elliptic). 18 November: Proposition 1: 'Viscosity comparison' (both viscosity sub/super-solution): First order: For nu u + H(x,Du)=0 with cont. cond. on H and nu > 0. (Proof: 'doubling the variables'). Proposition 2: 'Viscosity comparison' (both viscosity sub/super-solution): nu = 0: For H(x,Du) - f(x) = 0, homogeneity cond. on H and positivity cond. on f (+ cont. cond. on H); covers Eikonal equation. (Proof: Next time). Remarks: Includes Eikonal equations. 19 November: Proof of: Proposition (from last time): 'Viscosity comparison' (both viscosity sub/super-solution): nu = 0: For H(x,Du) - f(x) = 0, homogeneity cond. on H and positivity cond. on f (+ cont. cond. on H); covers Eikonal equation. Resume: Idea of proof of Comparison Principle (viscosity version) and 'Doubling of Variables'. Lemma: Resuming the desired properties of 'test function of double variables' (no proof). Discussion of literature (books). 02 December: Further discussion of literature: The methods just studied also works to prove Comparison Principle for the _Cauchy_ Problem for 1st order (degenerate) _parabolic_ equations. (For proof, see Achdou et al, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications p. 64.) Discussion: Why methods used for _1st_ order equations will not work for _2nd_ order equations: Example: nu u - Delta u = 0. Introduction/motivation: Semi-jets (1st and 2nd order). Definition of "little-o" - notation (Landau). Definition of (2nd order) semi-jets J^{2,+}u, J^{2,-}u. Remarks and observations. Proposition: (a) If, for some u, the intersection of the two (2nd order) semi-jets J^{2,+}u(x), J^{2,-}u(x) is non-empty at x, then Du(x) and D^2u(x) exist, and the pair (Du(x),D^2u(x)) is exactly this intersection. (b) The set of all points x in Omega for which the super-semi-jet J^{2,+}u(x) is non-empty is dense in Omega IF u is USC. (c) The set of all points x in Omega for which the sub-semi-jet J^{2,-}u(x) is non-empty is dense in Omega IF u is LSC. (Proof: Next time.) 09 December: Proof from last time (see above). Definition: "Closure" of semi-jets. Proposition: Equivalent definition of viscosity sub/super/solution via 2nd order semi-jets. 16 December: Corollary: Semi-jets characterised by derivatives of "allowed test functions". Example of semi-jets (of non-differentiable functions). Definition: Semi-jets, and their "closure", for functions defined on general (not necessarily open) sets. Remarks, examples. Proposition: Semi-jets of sum of two functions, one of which is C^2. Remarks, on literature. Recall: Discussion: Why methods used for _1st_ order equations will not work for _2nd_ order equations: Example: nu u - Delta u = 0. Overview of the plan for the alternative solution. Lemma: Ishii's Lemma/Theorem on Sums. (Proof: Possibly later!) Remarks. 17 December: Definition: The "Structure Condition" (for an equation F). Discussion. Proposition: (1) If F satisfies the Structure Condition, it is elliptic. (2) If F is uniformly elliptic, and satisfies an additional Lipschitz-like condition, then F satisfies the Structure Condition. (Proof: Possibly later!) Theorem: 'Viscosity Comparison' (both viscosity sub/super-solution): Second order: nu > 0 and "Structure Condition" on F. Theorem: 'Viscosity Comparison' (both viscosity sub/super-solution): Second order: nu >= 0, uniform ellipticity of F, uniform Lipschitz-cond. on F (in variable p), and "Structure Condition" on F (nu > 0 covered by previous theorem). NB Possible "preparation" for what follows: (Re-)Read "Perron's Method" (done in PDG1 (WiSe 2013/14): Pages 125-130 in: Q. Han und F. Lin, Elliptic Partial Differential Equations: Second Edition, AMS (Courant Lecture Notes), 2011. Pages 126-133 in: Q. Han, A Basic Course in Partial Differential Equations, AMS (Graduate Studies in Mathematics), 2011. Merry Christmas & Happy Newyear! 13 January: Chapter 3: Perron's Method. Discussion: For existence of solutions. Recall: Definition of semi-continuous envelopes. Lemma: Properties of semi-continuous envelopes. (Re-)Definition: Viscosity sub/super/solutions: If the corresponding semi-continuous envelopes are viscosity sub/super-solutions. Proposition: If the Comparison Principle holds, if u is a viscosity solutions, and if u^* = u_* on boundary Omega, then u is continuous in closure of Omega. Theorem: The pointwise supremum u (resp. infimum) of a non-empty set S of upper semi-continuous subsolutions (resp. lower semi-continuous supersolutions) is a viscosity subsolution (resp. supersolution) (in the sense just defined) IF the function u is bounded on compact sets and IF the F defining the equation is continuous. (NOTE: The oral exam takes place Friday 30 January 2015 - times to be determined). 20 January: Thereom (Existence of viscosity solutions: Perron's Method): For F continuous and degenerate elliptic, assume there exist viscosity subsolution xi, USC and locally bounded, and viscosity supersolution eta, LSC and locally bounded, of F = 0, with xi < = eta. Let S be the set of all viscosity subsolutions (of F = 0), USC and between xi and eta, and S^ the set of all viscosity supersolutions, LSC and between xi and eta. Let u be the pointwise sup over functions in S, and u^ the pointwise sup over functions in S^. Then u and u^ are viscosity _solutions_ of F = 0. Remarks and discissions. Proof (of Existence of viscosity solutions) (to be continued next time). 27 January: End of proof of Existence of viscosity solutions (from last time). Outlook (Koike's book): (a) Other existence results (1st order equations; via Dynamic Programming Principle). (b) Stability ('continuity in the data'). (c) Generalised Boundary Value Problem (BVP): Boundary condition 2nd order PDE (e.g. Dirichlet or Neumann), to hold in _viscosity_ sense. (d) L^p - viscosity solutions + regularity. (e) Appendix: Proof Ishii's Lemma, and lemmata needed for L^p-solutions, and regularity. End of class/course/lecture! Next semester: 'Viscosity Solutions for nonlinear PDEs 2', on regularity theory. ----------------------------------- Letzte Änderung: 29 January 2015 (no more updates). Thomas Østergaard Sørensen |
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