Hessian estimates for fully nonlinear equations via the large-\(M\)-inequality principle
type: Item
instance of: scholarly article
title: Hessian estimates for fully nonlinear equations via the large-\(M\)-inequality principle
zbMATH Open document ID: 1473.35230
author: Mikyoung Lee
author: Jihoon Ok
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-06-14T00:00:00Z
review text: The authors consider the fully nonlinear elliptic equation \(F(D^{2}u,x)=f(x)\) posed in an open subset \(\Omega \) of \(\mathbb{R}^{n}\) (\(n\geq 2\)). \( F:S\times \Omega \rightarrow \mathbb{R}\) is supposed to satisfy the uniform ellipticity property: \(\nu \left\Vert Y\right\Vert \leq F(X+Y,x)-F(X,x)\leq \Lambda \left\Vert Y\right\Vert \) a.e. \(x\in \Omega \) and for every \(X,Y\in S \) the space of real \(n\times n\) symmetric matrices, \(Y\) being positive semi-definite, among other hypotheses. The purpose of the paper is to prove a local \(W^{2,p}\)-estimate for viscosity solutions to this problem. The authors here use a large \(M\)-inequality proposed by \textit{E. Acerbi} and \textit{G. Mingione} in [Duke Math. J. 136, No. 2, 285--320 (2007; Zbl 1113.35105)] and they give another proof of the seminal regularity result by \textit{L.A. Caffarelli} in [Ann. Math. (2) 130, No. 1, 189--213 (1989; Zbl 0692.35017)]. They also complete this result with weighted Hessian estimates in variable exponent spaces for viscosity solutions, assuming that \(F\) is asymptotically convex. They first define the notion of \(L^{q}\)-viscosity solution (\(q>n/2\)) as a continuous function such that if \(F(D^{2}\varphi (x),x)-f(x)\geq +\epsilon \) and \(F(D^{2}\varphi (x),x)-f(x)\leq -\epsilon \) for every \(\varphi \in W_{\mathrm{loc}}^{2,q}(\Omega )\) and a.e. \(x\in \mathcal{O}\) an open subset of \(\Omega \) and \(\epsilon >0\), then \(u-\varphi \) cannot have a local minimum in \(\mathcal{O}\). The authors here assume that \(F\) is continuous in \(X\) and measurable in \(x\), and \(f\in L_{\mathrm{loc}}^{q}(\Omega )\). They introduce the recession function \(F^{\ast }\) associated with \(F\), variable exponents \(p(\cdot)\) satisfying \(n<p_{\Omega }^{-}=\inf_{x\in \Omega }p(x)\leq \sup_{x\in \Omega }p(x)=p_{\Omega }^{+}<+\infty \) and \(\left\vert p(x)-p(y)\right\vert \leq \theta (\left\vert x-y\right\vert )\) for \(x,y\in \Omega \) and where \(\theta :[0,+\infty )\rightarrow…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 42B37
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 7358799
zbMATH Keywords: fully nonlinear equation
zbMATH Keywords: viscosity solution
zbMATH Keywords: Hessian estimates
zbMATH Keywords: variable exponent space
zbMATH Keywords: Muckenhoupt weight
MaRDI profile type: Publication
full work available at URL: j.jmaa.2020.123953
OpenAlex ID: W3006526678
cites work: Gradient estimates for thep(x)-Laplacean system
cites work: Gradient estimates for a class of parabolic systems
cites work: \(W^{2,p(\cdot )}\)-regularity for elliptic equations in nondivergence form with BMO coefficients
cites work: Weighted regularity estimates in Orlicz spaces for fully nonlinear elliptic equations
cites work: Nondivergence parabolic equations in weighted variable exponent spaces
cites work: \(W^{2,p}\) estimates for solutions to asymptotically elliptic equations in nondivergence form
cites work: \(W^{1,p(\cdot)}\)-regularity for elliptic equations with measurable coefficients in nonsmooth domains
cites work: Elliptic equations with BMO coefficients in Reifenberg domains
cites work: Interior a priori estimates for solutions of fully nonlinear equations
cites work: Q4855688
cites work: On viscosity solutions of fully nonlinear equations with measurable ingredients
cites work: Q4389910
cites work: Interior Sobolev regularity for fully nonlinear parabolic equations
cites work: Calderón-Zygmund estimates and non-uniformly elliptic operators
cites work: User’s guide to viscosity solutions of second order partial differential equations
cites work: Borderline Estimates for Fully Nonlinear Elliptic Equations
cites work: A borderline case of Calderón–Zygmund estimates for nonuniformly elliptic problems
cites work: Lebesgue and Sobolev spaces with variable exponents
cites work: On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains
cites work: Q4287684
cites work: Classical solutions of fully nonlinear, convex, second-order elliptic equations
cites work: Modern Fourier Analysis
cites work: Elliptic partial differential equations
cites work: Q4750171
cites work: On the Existence of Solutions for Fully Nonlinear Elliptic Equations Under Relaxed Convexity Assumptions
cites work: On the existence of Wp2 solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions
cites work: Weighted Orlicz regularity estimates for fully nonlinear elliptic equations with asymptotic convexity
cites work: Second Derivative L p -Estimates for Elliptic Equations of Nondivergent Type
cites work: Q5428774
cites work: Sharp Hessian integrability estimates for nonlinear elliptic equations: an asymptotic approach
cites work: Regularity estimates for fully non linear elliptic equations which are asymptotically convex
cites work: Q3142876
cites work: \(W^{1,p}\)-interior estimates for solutions of fully nonlinear, uniformly elliptic equations
cites work: \(L^{p(\cdot), \lambda(\cdot)}\) regularity for fully nonlinear elliptic equations
cites work: Lorentz estimates for fully nonlinear parabolic and elliptic equations
cites work: On density of smooth functions in weighted Sobolev spaces with variable exponents
DOI: 10.1016/J.JMAA.2020.123953
description: scientific article; zbMATH DE number 7358799