**Mathematical Surveys and Monographs**

Volume: 233;
2018;
441 pp;
Hardcover

MSC: Primary 35;

**Print ISBN: 978-1-4704-4740-3
Product Code: SURV/233**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-4853-0
Product Code: SURV/233.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

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#### Supplemental Materials

# Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations

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*N. V. Krylov*

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.

#### Readership

Graduate students and researchers interested in nonlinear partial differential equations.

#### Reviews & Endorsements

The exposition is self-contained and extremely clear. This makes this book perfect for an advanced PhD class.

-- Vincenzo Vespri, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations

- Cover Cover11
- Title page i2
- Copyright ii3
- Contents iii4
- Preface vii8
- Bellman’s equations with constant “coefficients” in the whole space 116
- Estimates in 𝐿_{𝑝} for solutions of the Monge-Ampère type equations 1530
- The Aleksandrov estimates 3146
- First results for fully nonlinear equations 5570
- Finite-difference equations of elliptic type 7792
- Elliptic differential equations of cut-off type 101116
- Finite-difference equations of parabolic type 135150
- Parabolic differential equations of cut-off type 151166
- A priori estimates in 𝐶^{𝛼} for solutions of linear and nonlinear equations 165180
- Solvability in 𝑊²_{𝑝,𝑙𝑜𝑐} of fully nonlinear elliptic equations 207222
- Nonlinear elliptic equations in 𝐶^{2+𝛼}_{𝑙𝑜𝑐(Ω)∩𝐶(Ω)} 235250
- Solvability in 𝑊^{1,2}_{𝑝,𝑙𝑜𝑐} of fully nonlinear parabolic equations 251266
- Elements of the 𝐶^{2+𝛼}-theory of fully nonlinear elliptic and parabolic equations 271286
- Nonlinear elliptic equations in 𝑊²_{𝑝}(Ω) 311326
- Nonlinear parabolic equations in 𝑊^{1,2}_{𝑝} 329344
- 𝐶^{1+𝛼}-regularity of viscosity solutions of general parabolic equations 341356
- 𝐶^{1+𝛼}-regularity of 𝐿_{𝑝}-viscosity solutions of the Isaacs parabolic equations with almost VMO coefficients 367382
- Uniqueness and existence of extremal viscosity solutions for parabolic equations 381396
- Proof of Theorem 6.2.1 405420
- Proof of Lemma 9.2.6 409424
- Some tools from real analysis 413428
- Bibliography 431446
- Index 439454
- Back Cover Back Cover1458