The A. D. Aleksandrov maximum principle
type: Item
instance of: scholarly article
title: The A. D. Aleksandrov maximum principle
zbMATH Open document ID: 1157.35019
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-09-03T00:00:00Z
review text: The very interesting paper under review is devoted to the maximum principle of A. D. Aleksandrov which is one of the most beautiful applications of geometric ideas in the theory of the partial differential equations. After a short historical survey with rich reference links about the original result of Aleksandrov and its various outgrowths and applications, the author proposes some results related to the analysis of convex and convex-monotone holes. In particular, estimates for the maximum of solution to a Dirichlet problem for nondivergence form elliptic equations are derived. Moreover, local estimates are given for solutions to the degenerate Venttsel problem as well as to one- and two-phase Venttsel problems. A collection of the corresponding results for parabolic equations is also proposed. Highly recommended paper!
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35R05
zbMATH DE Number: 5319770
zbMATH Keywords: maximum principle
zbMATH Keywords: a priori estimates
zbMATH Keywords: nondivergence form elliptic and parabolic equations
zbMATH Keywords: equations with discontinuous data
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: degenerate Venttsel problem
zbMATH Keywords: two-phase Venttsel problems
author: A. I. Nazarov
MaRDI profile type: Publication
full work available at URL: s10958-007-0126-1
OpenAlex ID: W2073550784
cites work: Q5735387
cites work: Q5329326
cites work: Q5342616
cites work: Q5513924
cites work: Q5534503
cites work: Q4853573
cites work: Q4242545
cites work: Linear two-phase Venttsel problems
cites work: Q4391200
cites work: A survey of results on nonlinear Venttsel problems.
cites work: Elliptic second order equations
cites work: Interior a priori estimates for solutions of fully nonlinear equations
cites work: On the alexandroff‐bakelman‐pucci estimate and the reversed hölder inequality for solutions of elliptic and parabolic equations
cites work: Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations
cites work: The \(L^ p\)-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations
cites work: Maximum principle for second order elliptic equations and applications
cites work: On optimal estimates for some oblique derivative problems.
cites work: Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation
cites work: Q3728545
cites work: Q4492815
cites work: Maximum estimates for oblique derivative problems with right hand side in \(L^p\), \(p<n\)
cites work: An Aleksandrov-Bakelman type maximum principle and applications
cites work: Linear second order elliptic equations with Venttsel boundary conditions
cites work: Q3477251
cites work: Q3363014
cites work: Q4812155
cites work: Q3728446
cites work: Q5594015
cites work: Convex Analysis
cites work: Q3907899
cites work: Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations
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/S10958-007-0126-1
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description: scientific article; zbMATH DE number 5319770