On the Aleksandrov-Bakel'man-Pucci estimate for some elliptic and parabolic nonlinear operators
type: Item
instance of: scholarly article
title: On the Aleksandrov-Bakel'man-Pucci estimate for some elliptic and parabolic nonlinear operators
zbMATH Open document ID: 1257.35085
author: Roberto Argiolas
author: Fernando Charro
author: Ireneo Peral
published in: Archive for Rational Mechanics and Analysis
publication date: 2012-11-05T00:00:00Z
review text: The authors prove the Aleksandrov-Bakel'man-Pucci estimate for (possibly degenerate) nonlinear elliptic and parabolic equations of the form \[ -{\text{div}} (F(\nabla u(x))) = f(x) \quad {\text{in}}\,\Omega \subset \mathbb{R}^{n} \] and \[ u_{t}(x,t)-{\text{div}}(F(\nabla u(x,t))) = f(x,t) \quad {\text{in}}\,Q\subset \mathbb{R}^{n+1} \] for \(F\) a \({\mathcal{C}^1}\) monotone field under some suitable conditions. Examples of applications such as the \(p\)-Laplacian and the Mean Curvature Flow are considered, as well as extensions of the general results to equations that are not in divergence form, such as the \(m\)-curvature flow.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6101966
zbMATH Keywords: maximum principle
zbMATH Keywords: divergence form operators
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: s00205-011-0434-y
OpenAlex ID: W2049037908
cites work: The Dirichlet problem for singular fully nonlinear operators
cites work: On the alexandroff‐bakelman‐pucci estimate and the reversed hölder inequality for solutions of elliptic and parabolic equations
cites work: Q4855688
cites work: Interior a priori estimates for solutions of fully nonlinear equations
cites work: On viscosity solutions of fully nonlinear equations with measurable ingredients
cites work: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
cites work: Nonlinear second-order elliptic equations V. The dirichlet problem for weingarten hypersurfaces
cites work: Semiconcave functions, Hamilton-Jacobi equations, and optimal control
cites work: An Alexsandrov type theorem for k-convex functions
cites work: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations
cites work: Remarks on nonlinear uniformly parabolic equations
cites work: Lp- Theory for fully nonlinear uniformly parabolic equations
cites work: User’s guide to viscosity solutions of second order partial differential equations
cites work: Alexandroff-Bakelman-Pucci estimate for singular or degenerate fully nonlinear elliptic equations
cites work: Harnack inequality for singular fully nonlinear operators and some existence results
cites work: Degenerate parabolic equations
cites work: A Note on Krylov-Tso's Parabolic Inequality
cites work: Q4287684
cites work: Motion of level sets by mean curvature. I
cites work: The \(L^ p\)-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations
cites work: Q3377358
cites work: Q5187597
cites work: Convex hypersurfaces of prescribed curvatures
cites work: Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations
cites work: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations
cites work: A Uniqueness Result for Viscosity Solutions of Second Order Fully Nonlinear Partial Differential Equations
cites work: Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation
cites work: Q3909783
cites work: New maximum principles for linear elliptic equations
cites work: Viscosity supersolutions of the evolutionary \(p\)-Laplace equation.
cites work: On a generalization of Hadamard's determinantal inequality due to Szász
cites work: Singular degenerate parabolic equations with applications
cites work: Limitazioni per soluzioni di equazioni ellittiche
cites work: Q5486604
cites work: A priori bounds and necessary conditions for solvability of prescribed curvature equations
cites work: The Dirichlet problem for the prescribed curvature equations
cites work: Hessian measures. I
cites work: Hessian measures. II
cites work: On an Aleksandrov-Bakel'Man Type Maximum Principle for Second-Order Parabolic Equations
cites work: On the regularity theory of fully nonlinear parabolic equations: I
DOI: 10.1007/S00205-011-0434-Y
description: scientific article; zbMATH DE number 6101966