Vorlesung: Viscosity Solutions for nonlinear PDEs 2 (SoSe2015)

Vorlesung: Viscosity Solutions for nonlinear PDEs 2 (SoSe 2015)

Content of the lecture:

Literature referred to below:

[CC] L. A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations, AMS (Colloquium Publications), 1995.

[HL] Q. Han and F. Lin, Elliptic Partial Differential Equations: Second Edition, AMS (Courant Lecture Notes), 2011.

[J] J. Jost, Partial Differential Equations, Springer (Graduate Texts in Mathematics, Volume 214), 2013.

[K] S. Koike (Department of Mathematics, Saitama University, Japan), A Beginner’s Guide to the Theory of Viscosity Solutions, 2nd edition (version: June 28, 2012).

[GT] D. Gilbarg und N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag (Classics in Mathematics Volume 224), 2001.

[WKK] E. Wienholtz, H. Kalf, T. Kriecherbauer, Elliptische Differentialgleichungen zweiter Ordnung, Springer, 2009.

[Alt] H. W. Alt, Lineare Funktionalanalysis, Springer (Springer-Lehrbuch Masterclass), 2012.

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

[G1] Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser (ETH Zürich Lectures in Mathematics), 1993.

[G2] Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, AMS (Annals of Mathematics Studies 105), 1983.

[P] Peetre, On convolution operators leaving L^{p,lambda}-spaces invariant, Annali di Matematica Pura ed Applicata, Volume 72, Issue 1, pp 295-304, 1966.

[T] Trudinger, A new approach to the Schauder estimates for linear elliptic equations, Proc. Centre Math. Appl., 1986.

[B] C. Baker, A proof of the parabolic Schauder estimates using Trudinger's method and the mean value property of the heat equation, arXiv:1204.0882 [math.AP], 2012.

[S] L. Simon, Schauder estimates by scaling Calc of Var's and PDE, Volume 5, Issue 5, pp. 391-407, 1997.

[CW] Y.-Z. Chen, L.-C. Wu, Second Order Elliptic Equations and Elliptic Systems, AMS (Translations of Mathematical Monographs), 2004.

[BCESS] M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner, P. E. Souganidis, Viscosity Solutions and Applications, Springer (LNM 1660), 1997.

[A] L. Ambrosio, Lecture Notes on Elliptic Partial Differential Equations, PhD course (given in 2009-2010 and then in 2012-2013, 2014-2015, lectures typed by A.Carlotto and A.Massaccesi).

[H] P. Hintz, Viscosity Solutions, Notes for a talk in the Student Analysis and Geometry Seminar, Stanford University (April 25th, 2014).

[R] M. Rang, Regularity Results for Nonlocal Fully Nonlinear Elliptic Equations, Dissertation, Bielefeld University, 2013.

[Ka] M. Kassmann, Harnack Inequalities: An Introduction, Boundary Value Problems, Volume 2007, Article ID 81415, 2006.

[G] C. E. Gutierrez, The Monge-Ampère Equation, Birkhäser (Progress in Nonlinear Differential Equations and Their Applications), 2001.

14 April:

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). Linear equations in _divergence_ and _non-divergence_ form. Discussion: 'weak solutions': For _divergence_ form (mostly). 'Viscosity solutions': For non-divergence - but especially for _really_ ('fully') nonlinear.

Examples: Linear (non-divergence form), 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 mention Dirichlet Boundary Value Problem (BVP): Value on boundary of domain is given.

Discussion/motivation: (Hadamard) Well-posedness: Existence, uniqueness, and stability/continuity in data. Difficulty: Existence of _classical_ solutions.

21 April:

Approach: Define 'useful' concept of 'generalised solution' (in PDE2/for 2nd order linear uniformly elliptic eq's in div form: weak derivatives, Sobolev 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 last course. This course: regularity.

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.

A (brief!) motivation for the _definition_ of 'viscosity solution': Using the Maximum Principle on classical sub/super/solution of -Delta u = 0 (or more generally of -tr(A D^2 u) = 0, A >= 0). For more motivation, see previous course.

Definition of viscosity subsolution/supersolution/solution.

28 April:

Remarks: On assumptions.

Proposition: Equivalent formulation of the definition of viscosity sub/super/solution: Via testfunctions phi such that u-phi attains strict max, which is 0.

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.

General remarks: On regularity studies, and a priori estimates (quantative statement, instead of only qualitative statement).

Definition: Hölder spaces.
Theorem: Hölder spaces are Banach; compact embeddings (no proof). Discussions.

General 'plan' for regularity studies: C^alpha, C^{1,alpha}, and C^{2,alpha}-estimates (for _weak_ solutions: Need to start with L-infinity estimates). Each (often) under condition that have previous one. Each under certain conditions on the equation.

05 May:

The equations we will study: On form F(x,D^2u)=f(x) - as perturbation of F(D^2u)=0.
Discussion of a priori estimates: Interior, boundary, and global estimates (respectively, regularity).

Chapter 1: Schauder estimates, C^{2,alpha}-estimates, and The Continuity Method.

Also called 'Method of Continuity'.

Will give another reason to study global C^{2,alpha}-estimates.

Theorem: L_t norm-continuous family of bounded operators from B (Banach) to V (normed), for t in [0,1]. Assume lower bound on L_t(x), uniform in x and t. Then L_0 is surjective iff L_1 is surjective. (No proof).

Use: Prove existence of solutions via Method of Continuity.

Theorem: Let L be a uniformly elliptic 2nd order linear operator on a bounded C^{2,alpha}-domain, with C^alpha coefficients. If the Dirichlet BVP for the Poisson equation ( -Delta u = f) on the domain has a C^{2,alpha}-solution for any right hand side f in C^alpha and boundary value phi in C^{2,alpha}, then the Dirichlet BVP for the operator L also has a (unique) C^{2,alpha}-solution for any such f & phi. (End of proof next time.)
See [HL] pp. 134-136.

Proof based on:

Theorem (Schauder estimates): Global a priori C^{2,alpha}-estimate for C^{2,alpha}-solutions to uniformly elliptic 2nd order linear equation with C^alpha-coefficients, C^alpha right hand side f, and C^{2,alpha} boundary value phi. (Proof to be discussed later.)

12 May:

End of proof (from last time): C^{2,alpha}-solvability of Dirichlet BVP for general uniformly elliptic 2nd order linear operator with C^alpha-coefficients via Method of Continuity.

Discussion and remarks.

Theorem: C^infinity solvability of uniformly elliptic concave (or convex) fully nonlinear Dirichlet BVP F(D^2u)=0, and C^{2,alpha}-estimate.
(NO proof; is by Method of Continuity and global a priori C^{2,alpha}-estimates, see [CC] pp. 95-97.)

Comment on Method of Continuity: Yau's proof of the Calabi Conjecture.

Theorem: Global a priori C^{2,alpha}-estimate for solutions to uniformly elliptic concave (or convex) fully nonlinear Dirichlet BVP F(D^2u)=0.
(NO proof here; [CC] Thm 9.5 p. 88.)

Two applications of Schauder estimates to _nonlinear_ equations:
(a) Higher regularity for fully nonlinear uniformly elliptic 2nd order equations.
(b) Existence for BVP for (certain) quasi-linear equations.

(a) Theorem: If u is a C^{2,alpha}-solution to uniformly elliptic equation F(x,D^2u)=f(x), with F & f C^infinity, then u is C^infinity. (End of proof next time.)

19 May:

End of proof from last time (see above). See [CC] pp. 85-86.

(b) Existence for BVP for (certain) quasi-linear equations: Via the Leray-Schauder Fixpoint Theorem, for uniformly elliptic linear equations with C^alpha-coefficients. Uses: Existence of solution to the _linear_ problem, and an a priori C^{1,alpha}-estimate on such solutions.
See [HL] pp. 142-144.

On the proof of Schauder estimates:
Tactic: (a) Do for Laplace, (b) for constant coefficients, (c) for C^alpha-coefficients (by perturbation-argument / "freezing the coefficients").

Methods to prove Schauder:

(1) Potential theory (more next time).
See f.ex. [J] pp. 329-346. See also [WKK].

02 June:

Methods to prove Schauder (continued):

(1) Potential theory (continued) and singular integral operators, on Hölder spaces and L^p-spaces (without proofs). See [Alt] pp. 321-325; 330; 339-350.
(Comments on additional literature on Singular Integral Operators, Harmonic Analysis, and Calderon-Zygmund Operators.)

(2) Campanato spaces: Definition and properties of Morrey and Campanato spaces (without proofs). Gives equivalent characterization of Hölder spaces via integral inequalities.
See [G1], [G2], and [HL] Chapter 3.

09 June:

Methods to prove Schauder (continued):

(3) Convolution operators on Campanato-spaces. See [P].

(4) Mollification: The mollification of a solution satisfies a mollified equation which can be differentiated three times. Using the mean value inequality for subharmonic functions gives an estimate on the third derivative of the mollification of the solution. One gets Schauder by using an equivalent description of Hölder-continuity of v via the derivative of the mollification of v.
See [T], [B], [CW].

(5) Via scaling. See [S].

Resume of Chapter 1, including statement of 'Evans-Krylov Theorem': C^{2,alpha}-regularity of solutions to convex uniformly elliptic fully nonlinear equation (with 'constant coefficients') F(D^2u)=0.

Chapter 2: Pucci's extremal operators and their equations.

Definition: Affine functions, paraboloids, paraboloids of opening M, u touching v from above at x_0 in set A.

Recall: Definition of 'elliptic' and of 'viscosity sub-solution/super-solution/solution'.

23 June:

Recall (Visc Sol 1, Prop. 1.10; [K] Prop. 2.4):
Proposition: If u is a (USC) viscosity subsolution (resp. (LSC) 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.
Corollary: If u is a (continous) viscosity solution on Omega, bounded on Omega, then it is a viscosity solution on any open subset Omega' of Omega.

Discussion of regularity (Hölder, gradient-Hölder, C^{2,alpha}) as question of how well can approximate function locally by nice functions (constant, affine, paraboloids).

Definition: Pucci's extremal operators (P^{+}(X), P^{-}(X)).

Remarks: Equivalent definition (incl proof): Via eigenvalues of X.
Connection to Pucci's maximal operators (M^{+}(X), M^{-}(X)).

Lemma: Various properties of Pucci's operators.

Definition: Uniform ellipticity of fully nonlinear equation F (via Pucci's operators).
Remarks: As consequence, F is Lipschitz in X.
[CC] p. 12, 14-15; [HL] p. 100-101; [K] p. 24-25.

24 June:

Further remarks on uniform ellipticity: As a consequence, the 'linearization' of a uniformly elliptic F (about any matrix X) is a linear uniformly elliptic _linear_ operator (in the usual sense).

Definition: Pucci's equations, and the classes S_, S^{-}, S, S^*.

Lemma: Various properties of S^# (S^# = S_, S ^{-}, S, S^*), including scaling & translation (more added 30 June).

Proposition: Any subsolution of a uniformly elliptic equation (fully nonlinear) is a subsolution of _some_ Pucci equation (similarly for supersolution).
Consequence: Every (!?) regularity result for Pucci equations transfers to uniformly elliptic equations.

Remark: The class S (somehow) contains 'all solutions to all uniformly elliptic linear equations with same ellipticity contants'.

[BCESS] p. 99-100; [CC] p. 15-16; [HL] p. 100-101.

30 June:

Added to: Lemma: Various properties of S^# (S^# = S_, S ^{-}, S, S^*), including scaling & translation.

Remark: If u is in S^#, then so is u+c for any constant c.

Chapter 3: Aleksandrov-Bakelman-Pucci (ABP) estimate, (ABP Max Principle), Harnack inequality, and Hölder continuity (C^alpha).

Goal: Understand why Harnack implies C^alpha. Possibly, prove/indicate proof of Harnack.

Other cases of C^alpha (from Harnack): Uniformly elliptic linear equations with bounded coefficients: Divergence form (weak solutions): De Giorgi-Nash-Moser (no proof!). Non-divergence form (strong solutions): Krylov-Safonov (no proof!).

Recall: Harnack, for harmonic functions, and, more generally, for _classical_ solutions to uniform elliptic linear equations (no proof). (See [E] p. 351; [GT]).

Remark: Harnack also valid/important for _parabolic_ equations.

Theorem: Harnack for u in S^*(f) in cube Q_1, with f continuous and bounded. (NO PROOF YET!)

Corollary: Scaled version of Harnack, on cubes Q_R(x). (Proof).

[HL] p. 104-105 (see also [HL] p. 78-92). [CC] p. 31; 37-38.

See also (also for next two lectures/Chapter 2 above): [A] Chapter 20 + 21 (but also other parts); [H]; [R] p. 9-37.
For everything (!!) on Harnack inequalities, see [Ka].

07 July:

Theorem: Harnack implies C^{alpha} (Hölder)
(if the equation satisfies scaling/translation & that u + c is solution if u is solution):
(1) Oscillation estimate.
(2) A priori (interior) C^alpha-estimate.
(Including proof). (See also [R] Prop 2.32; [HL] Thm 5.10; [CC] Prop 4.10; [A] p. 160-161; [H] Cor 3.2.)

Remarks; in particular: Technical 'Iteration' Lemma ([HL] Lemma 4.19 f.ex.) (no proof).
Harnack will follow from:
Proposition: There exist epsilon and C such that: If inf u over Q_{1/4} is <= 1 for some u in S^*(f) with |f|_{L^n} smaller than epsilon then sup u over Q_{1/4} is bounded by C.

Proof that Proposition implies Harnack.
(See also [R] Lem 2.26 (all details of proof!); [HL] Lem 5.12; [CC] Lem 4.4; [A] Thm 21.9 (incl proof); [H] Lem 3.3.)

Strategy for proof of Proposition:
(1) A decay estimate on 'level sets' (using u in S^{-}(|f|)).
(2) Prove the bound in Proposition by decay estimate _and_ using that u is in S_(-|f|).
(To be continued next time.)

14 July:

Lemma: Decay of (Lebesque measure of) (super)level sets (of positive super-solutions).
('If us small somewhere in large cube Q_3 then it can be bounded on a large subset of Q_1').
Follows (proof given) from more 'technical' lemma of same type (but 'discrete').

The proof of this lemma: 3 main ingredients.

First ingredient:
Lemma: Dyadic Lemma / Calderon-Zygmund Decomposition (of cubes). (No proof; definition of 'dyadic cubes' given).
Consequence of more general Calderon-Zygmund Decomposition of functions (on cubes). (No proof).

Second ingredient:
Lemma: Existence of a 'barrier function' with certain particular properties. (No proof).

Third - and most important - ingredient (the one that _uses_ the equation): A _quantitative_ max principle.
Definition and notation: Convex/concave envelopes, lower and upper contact sets.
Theorem: Classical (C^2) version of Alexandrov-Bakelman-Pucci (ABP) Maximum Principle: _Quantitative_ estimate of how far a C^2 function is from being convex. (No proof; main ingredient: The Area Formel; does not use any equation).
Definition: Gamma_u: The lower contact set of minus the extension (by zero) to B_{2r} (from B_r) of u^{-} (!)
Theorem (weaker ABP): For continuous non-negative u, if Gamma_u is C^{1,1}, the ABP estimate holds. (No proof; does not use any equation.)
The theorem which is the 'third ingredient' (in proof of 'technical' lemma mentioned above): DOES use equation:
Theorem (ABP weak max principle): Assume u in S^*(f) (with f continuous on closed ball), non-negative, and continuous on closed ball. Then ABP estimate holds. (No proof; indication what needed to prove to use previous thm.)

Brief discussion how three main ingredients lead to proof of 'technical' lemma (decay of superlevels).
Resume of chain of implications that lead to proof of Harnack, and then of Hölder continuity.

See [A] p. 154-169; [R] p. 25-33; [HL] p. 37-40, 102-109; [CC] p. 21-28, 29-37; [H] p. 3-10.

For a proof of the Krylov-Safonov Theorem (Solutions to _linear_ uniformly elliptic eq's in non-divergence form with bounded coeff's are Hölder) along the same lines, see [G] p. 12-17, 31-43.

End of Lectures!

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Letzte Änderung: 15 July 2015 (no more updates).

Thomas Østergaard Sørensen

Teaching