The behavior of weak solutions to the boundary value problems for elliptic quasilinear equations with triple degeneration in a neighborhood of a boundary edge.
type: Item
instance of: scholarly article
title: The behavior of weak solutions to the boundary value problems for elliptic quasilinear equations with triple degeneration in a neighborhood of a boundary edge.
zbMATH Open document ID: 1100.35035
author: M. V. Borsuk
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-03-14T00:00:00Z
review text: The author investigates the behaviour of solutions to boundary value problems (Dirichlet, Neumann, mixed) for the equation \[ -\frac{d}{dx_i}a_i(x,u,u_x)+a(x,u,u_x)+b(x,u,u_x)=f(x) \] in a domain \(G\subset \mathbb R^n\), \(n\geq 3\). The coefficients \(a_i(x,u,u_x)\) satisfy a degenerate ellipticity condition: a typical example is \[ a_i(x,u,u_x)= (x^2_{n-1}+x^2_n)^\frac{\tau}{2}| u| ^q| \nabla u| ^{m-2}u_{x_i}, \] \(q\geq 0\), \(m\geq 2\), \(\tau\geq m-2\). The functions \(a(x,u,u_x)\), \(b(x,u,u_x)\) and \(f(x)\) satisfy suitable restrictions. The domain \(G\) possesses an edge, and the paper is devoted mainly to the estimate of the weak solutions to the boundary value problems near this edge. Suitable weighted Sobolev spaces are introduced and an almost sharp estimate of \(u(x)\) in a neighborhood of the edge is given.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 2056129
zbMATH Keywords: degenerate quasilinear elliptic equations
zbMATH Keywords: edges
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Neumann problem
zbMATH Keywords: mixed problem
reviewed by: Giovanni Porru
MaRDI profile type: Publication
full work available at URL: j.na.2003.09.012
OpenAlex ID: W1979799375
cites work: Degenerate elliptic operators diagonal systems and variational integrals
cites work: Q4846336
cites work: Q4881218
cites work: On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points
cites work: Q4513893
cites work: \(L^ \infty\)-solutions for some nonlinear degenerate elliptic and parabolic equations
cites work: On quasilinear elliptic equations in domains with conical boundary points.
cites work: Q4343193
cites work: Elliptic partial differential equations of second order
cites work: Q3470840
cites work: Linear and quasilinear elliptic equations
cites work: Boundary value problems for some degenerate-elliptic operators
cites work: On The Dirichletproblem for Quasilinear Equations
DOI: 10.1016/J.NA.2003.09.012
description: scientific article; zbMATH DE number 2056129