Summability properties of solutions of second-order elliptic equations
type: Item
instance of: scholarly article
title: Summability properties of solutions of second-order elliptic equations
zbMATH Open document ID: 0656.35034
DOI: 10.1007/BF01138841
published in: Mathematical Notes
publication date: 1988-01-01T00:00:00Z
review text: The authors consider the Dirichlet problem \[ \sum^{n}_{i,j=1}(\partial /\partial x_ i)(a_{ij}\partial u/\partial x_ j)+V(x)u+\lambda u=f,\quad u|_{\partial \Omega}=0 \] where the function V satisfies the following condition: If for any \(\epsilon >0\) there exists c(\(\epsilon)\) such that \[ -\int_{\Omega}V^ 2_{\phi} dx\leq \epsilon \int_{\Omega}| {\bar \nabla}\phi |^ 2 dx+c(\epsilon)\int_{\Omega}\phi^ 2 dx \] for each \(\phi\) belonging to \(C_ 0^{\infty}(\Omega).\) Under some additional conditions on V the authors obtain two estimates of the solution to the problem. In particular, the authors prove the estimate of the weak solution \[ \int_{\Omega}| x|^{-2} | u(x)|^ q dx\leq K \exp (qj/(j+q))\int^{1/2}_{j/(q+2j)}t^{-2} \ln [t^{-1}c(t)]dt. \] If u is a weak solution to the Dirichlet problem for an equation \(\Delta u=f\) in \(L^ 2\) then they prove an estimate \(\int_{\Omega} | x|^{-2} (e^{\beta | u|}- 1)dx<\infty,\) \(\forall \beta <(n-2)^ 2\pi^{n/2}/(\Gamma (1+n/2)\| f\|_{n/2},\omega)^ 3,\quad f\in L_{\omega}^{n/2}.\)
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4071442
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: estimates
zbMATH Keywords: weak solution
author: V. A. Liskevich
author: Michael Perelmuter
reviewed by: Leonid Lebedev
MaRDI profile type: Publication
cites work: Q5547919
cites work: Q5651750
cites work: Q5614914
cites work: Q3344520
cites work: Q4765475
full work available at URL: https://doi.org/10.1007/bf01138841
OpenAlex ID: W2047064562
Recommended article: Summability for solutions to some quasilinear elliptic systems
Recommended article: On a differential property of solutions of second-order elliptic equations
Recommended article: Q5609947
Recommended article: Q3743679
Recommended article: Optimal summability of solutions to nonlinear elliptic problems
Recommended article: Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation
Recommended article: Q3808528
Recommended article: Q4338517
Recommended article: On the summability in \(L^ p\) of positive solutions of elliptic equations.
Recommended article: Convexity properties for solutions of some second order elliptic semilinear equations
description: scientific article; zbMATH DE number 4071442