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# Sobolev and viscosity solutions for fully nonlinear elliptic and parabolic equations

Author: N V Krylov Providence, Rhode Island : American Mathematical Society, [2018] ©2018 Mathematical surveys and monographs, no. 233. Print book : EnglishView all editions and formats "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.  Read more... (not yet rated) 0 with reviews - Be the first.

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Document Type: Book N V Krylov Find more information about: N V Krylov 9781470447403 1470447401 1039672482 "Applied mathematics"--Cover. xiv, 441 pages ; 27 cm. Bellman's equations with constant coefficients'' in the whole spaceEstimates in $L_p$ for solutions of the Monge-Ampere type equationsThe Aleksandrov estimatesFirst results for fully nonlinear equationsFinite-difference equations of elliptic typeElliptic differential equations of cut-off typeFinite-difference equations of parabolic typeParabolic differential equations of cut-off typeA priori estimates in $C^\alpha$ for solutions of linear and nonlinear equationsSolvability in $W^2_{p,\mathrm{loc}}$ of fully nonlinear elliptic equationsNonlinear elliptic equations in $C^{2+\alpha}_{\mathrm{loc}}(\Omega)\cap C(\overline{\Omega})$Solvability in $W^{1,2}_{p,\mathrm{loc}}$ of fully nonlinear parabolic equationsElements of the $C^{2+\alpha}$-theory of fully nonlinear elliptic and parabolic equationsNonlinear 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 coefficientsUniqueness and existence of extremal viscosity solutions for parabolic equationsAppendix A. Proof of Theorem 6.2.1Appendix B. Proof of Lemma 9.2.6Appendix C. Some tools from real analysisBibliographyIndex Mathematical surveys and monographs, no. 233. N.V. Krylov.

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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  Read more...

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