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Hardcover ISBN:  9781470447403 
Product Code:  SURV/233 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470448530 
Product Code:  SURV/233.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470447403 
eBook ISBN:  9781470448530 
Product Code:  SURV/233.B 
List Price:  $254.00 $191.50 
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AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 233; 2018; 441 ppMSC: Primary 35
This book concentrates on first boundaryvalue problems for fully nonlinear secondorder 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 FangHua Lin's like estimates, and on the socalled “ersatz” existence theorems, saying that one can slightly modify “any” equation and get a “cutoff” 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.
ReadershipGraduate students and researchers interested in nonlinear partial differential equations.

Table of Contents

Chapters

Bellman’s equations with constant “coefficients” in the whole space

Estimates in $L_p$ for solutions of the MongeAmpère type equations

The Aleksandrov estimates

First results for fully nonlinear equations

Finitedifference equations of elliptic type

Elliptic differential equations of cutoff type

Finitedifference equations of parabolic type

Parabolic differential equations of cutoff type

A priori estimates in $C^\alpha $ for solutions of linear and nonlinear equations

Solvability in $W^2_{p,loc}$ of fully nonlinear elliptic equations

Nonlinear elliptic equations in $C^{2+\alpha }_{loc(\Omega )\cap C(\bar \Omega )}$

Solvability in $W^{1,2}_{p,loc}$ of fully nonlinear parabolic equations

Elements of the $C^{2+\alpha }$theory of fully nonlinear elliptic and parabolic equations

Nonlinear elliptic equations in $W^2_p(\Omega )$

Nonlinear parabolic equations in $W^{1,2}_p$

$C^{1+\alpha }$regularity of viscosity solutions of general parabolic equations

$C^{1+\alpha }$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


Additional Material

Reviews

The exposition is selfcontained and extremely clear. This makes this book perfect for an advanced PhD class.
Vincenzo Vespri, Zentralblatt MATH


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This book concentrates on first boundaryvalue problems for fully nonlinear secondorder 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 FangHua Lin's like estimates, and on the socalled “ersatz” existence theorems, saying that one can slightly modify “any” equation and get a “cutoff” 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.
Graduate students and researchers interested in nonlinear partial differential equations.

Chapters

Bellman’s equations with constant “coefficients” in the whole space

Estimates in $L_p$ for solutions of the MongeAmpère type equations

The Aleksandrov estimates

First results for fully nonlinear equations

Finitedifference equations of elliptic type

Elliptic differential equations of cutoff type

Finitedifference equations of parabolic type

Parabolic differential equations of cutoff type

A priori estimates in $C^\alpha $ for solutions of linear and nonlinear equations

Solvability in $W^2_{p,loc}$ of fully nonlinear elliptic equations

Nonlinear elliptic equations in $C^{2+\alpha }_{loc(\Omega )\cap C(\bar \Omega )}$

Solvability in $W^{1,2}_{p,loc}$ of fully nonlinear parabolic equations

Elements of the $C^{2+\alpha }$theory of fully nonlinear elliptic and parabolic equations

Nonlinear elliptic equations in $W^2_p(\Omega )$

Nonlinear parabolic equations in $W^{1,2}_p$

$C^{1+\alpha }$regularity of viscosity solutions of general parabolic equations

$C^{1+\alpha }$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

The exposition is selfcontained and extremely clear. This makes this book perfect for an advanced PhD class.
Vincenzo Vespri, Zentralblatt MATH