On doubling properties for non-negative weak solutions of elliptic and parabolic PDE
type: Item
instance of: scholarly article
title: On doubling properties for non-negative weak solutions of elliptic and parabolic PDE
zbMATH Open document ID: 0942.35093
DOI: 10.1007/BF02810591
author: Kazuhiro Kurata
published in: Israel Journal of Mathematics
publication date: 2000-06-07T00:00:00Z
review text: The paper studies quantitative properties for non-negative weak supersolutions for parabolic and elliptic PDE \[ \partial_t u-\operatorname {div}{\mathcal A}(t,x,u,\nabla u)+{\mathcal B} (t,x,u,\nabla u)=0\quad \text{in }(0,T)\times \Omega \] and \[ -\operatorname {div}{\mathcal A}(x,u,\nabla u)+{\mathcal B} (x,u,\nabla u)=0\quad \text{in }\Omega, \] where \(\Omega\subset{\mathbb R}^n\), \(n\geq 3.\) The doubling property of \(u^\delta\) with some small exponent \(1>\delta >0\) is established for these solutions instead of Harnack's inequality when the coefficients have strong singularity. Furthermore, the doubling property of \(u^q\) with large exponent \(2(n+2)/n\geq q>0\) for non-negative weak solutions \(u\) of the parabolic equation is shown.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 1436109
zbMATH Keywords: weak supersolutions
zbMATH Keywords: Harnack inequality
reviewed by: Lubomira Softova
MaRDI profile type: Publication
cites work: Local behavior of solutions of quasilinear parabolic equations
cites work: Harnack's Inequality for Schrodinger Operators and the Continuity of Solutions
cites work: A Harnack inequality for degenerate parabolic equations
cites work: A remark on a Harnack inequality for dengerate parabolic equations
cites work: LPEstimates for fractional integrals and sobolev inequalities with applications to schrödinger operators
cites work: The local regularity of solutions of degenerate elliptic equations
cites work: Q3834387
cites work: Unique continuation for elliptic operators: A geometric-variational approach
cites work: Q3700300
cites work: Q4314549
cites work: Q4309714
cites work: A unique continuation theorem for uniformly elliptic equations with strongly singular potentials
cites work: A harnack inequality for parabolic differential equations
cites work: Q5286841
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cites work: Weighted Norm Estimates for Sobolev Spaces
cites work: Q3756738
cites work: Recent work on sharp estimates in second-order elliptic unique continuation problems
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description: scientific article; zbMATH DE number 1436109