Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations (Mathematical Surveys and Monographs) (Mathematical Surveys and Monographs, 233)

This book concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. We look for solutions in Sobolev classes, local or global, or for viscosity solutions. Most of the auxiliary results, such as Aleksandrov's elliptic and parabolic estimates, the Krylov Safonov and the Evans Krylov theorems, are taken from old sources, and the main results were obtained in the last few years. Presentation of these results is based on a generalization of the Fefferman Stein theorem, on Fang-Hua Lin's like estimates, and on the so-called ersatz existence theorems, saying that one can slightly modify any equation and get a cut-off equation that has solutions with bounded derivatives. These theorems allow us to prove the solvability in Sobolev classes for equations that are quite far from the ones which are convex or concave with respect to the Hessians of the unknown functions. In studying viscosity solutions, these theorems also allow us to deal with classical approximating solutions, thus avoiding sometimes heavy constructions from the usual theory of viscosity solutions. Cover Title page Copyright Contents Preface Bellman’s equations with constant “coefficients” in the whole space Estimates in L_{p} for solutions of the Monge-Ampère type equations The Aleksandrov estimates First results for fully nonlinear equations Finite-difference equations of elliptic type Elliptic differential equations of cut-off type Finite-difference equations of parabolic type Parabolic differential equations of cut-off type A priori estimates in C^{α} for solutions of linear and nonlinear equations Solvability in W2_{p,loc} of fully nonlinear elliptic equations Nonlinear elliptic equations in C^{2+α}_{loc(Ω)∩C(Ω)} Solvability in W^{1,2}_{p,loc} of fully nonlinear parabolic equations Elements of the C^{2+α}-theory of fully nonlinear elliptic and parabolic equations Nonlinear elliptic equations in W2_{p}(Ω) Nonlinear parabolic equations in W^{1,2}_{p} C^{1+α}-regularity of viscosity solutions of general parabolic equations C^{1+α}-regularity of L_{p}-viscosity solutions of the Isaacs parabolic equations with almost VMO coefficients Uniqueness and existence of extremal viscosity solutions for parabolic equations Proof of Theorem 6.2.1 Proof of Lemma 9.2.6 Some tools from real analysis Bibliography Index Back Cover "This book concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. We look for solutions in Sobolev classes, local or global, or for viscosity solutions. Most of the auxiliary results, such as Aleksandrov's elliptic and parabolic estimates, the Krylov-Safonov and the Evans-Krylov theorems, are taken from old sources, and the main results were obtained in the last few years. Presentation of these results is based on a generalization of the Fefferman-Stein theorem, on Fang-Hua Lin's like estimates, and on the so-called "ersatz" existence theorems, saying that one can slightly modify "any" equation and get a "cut-off" equation that has solutions with bounded derivatives. These theorems allow us to prove the solvability in Sobolev classes for equations that are quite far from the ones which are convex or concave with respect to the Hessians of the unknown functions. In studying viscosity solutions, these theorems also allow us to deal with classical approximating solutions, thus avoiding sometimes heavy constructions from the usual theory of viscosity solutions."--Back cover Concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. The authors look for solutions in Sobolev classes, or for viscosity solutions. Most of the auxiliary results are taken from old sources, and the main results were obtained in the last few years.