Potential Analysis

, Volume 47, Issue 1, pp 21–36 | Cite as

Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure

Article

Abstract

We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of
$${\int}_{\Omega} f(x,u,\nabla u)\,dx $$
with f subject to the general structural conditions
$$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$
where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, gLt(Ω) for some t > n/p, and the lower weak Harnack estimate is proved under the stronger assumption that b, gL(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, gL(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Hölder-continuous.

Keywords

Calculus of variations De Giorgi estimate Harnack inequality Lebesgue points Nonstandard growth p-Laplace p(⋅)-Laplacian Quasiminimizers Variable exponent Variational integral 

Mathematics Subject Classification (2010)

49N60 35J20 35J62 

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland
  2. 2.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland

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