An Aleksandrov-Bakelman type maximum principle and applications
type: Item
instance of: scholarly article
title: An Aleksandrov-Bakelman type maximum principle and applications
zbMATH Open document ID: 0812.35014
author: Yousong Luo
published in: Journal of Differential Equations
publication date: 1993-05-16T00:00:00Z
review text: This paper is concerned with various local estimates of solutions of second order linear elliptic equations with Venttsel boundary conditions. Venttsel boundary conditions arose from probability theory [\textit{A. D. Venttsel}, Theor. Probab. Appl. 4, 164-177 (1960), transl. from Teor. Verojatn. Primen. 4, 172-185 (1959; Zbl 0089.134)] and three-dimensional water wave theory [\textit{M. Shinbrot}, J. Inst. Math. App. 25, 367-385 (1980; Zbl 0441.76009)]. The existence and uniqueness of classical solutions to linear Venttsel boundary value problems is treated by \textit{Y. Luo} and \textit{N. S. Trudinger} [Proc. R. Soc. Edinb., Sect. A 118, No. 3/4, 193-207 (1991; Zbl 0771.35014)]. The work described in this paper is preliminary for the study of the corresponding nonlinear Venttsel boundary value problems [\textit{Y. Luo}, Nonlinear Anal., Theory Methods Appl. 16, No. 9, 761-769 (1991; Zbl 0731.35035), \textit{Y. Luo} and \textit{N. S. Trudinger}, Potential Anal. 3, No. 2, 219-243 (1994)]. The existence result for nonlinear problems usually depends strongly upon a priori estimates. In Venttsel boundary value problems the boundary Hölder estimates are desired, since the interior estimates are known. This paper gives a boundary version of the Aleksandrov-Bakelman maximum principle and a boundary version of the weak Harnack inequality which are fundamental for the boundary Hölder estimates for solutions to nonlinear Venttsel boundary value problems.
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 166899
zbMATH Keywords: Venttsel boundary conditions
zbMATH Keywords: boundary Hölder estimates
zbMATH Keywords: Aleksandrov- Bakelman maximum principle
zbMATH Keywords: weak Harnack inequality
reviewed by: Shigeru Sakaguchi
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1011
OpenAlex ID: W2067423211
DOI: 10.1006/JDEQ.1993.1011
description: scientific article; zbMATH DE number 166899