Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth

We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, Δ<inf>p(x)</inf>u := div(|∇<inf>u</inf>|p(x)-2∇<inf>u</inf>) = f(x) in Ω fo...

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Autor principal: Wolanski, N.
Formato: JOUR
Materias:
Harnack's inequality
Local bounds
Variable exponent spaces
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00416932_v56_n1_p73_Wolanski
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00416932_v56_n1_p73_Wolanski2023-10-03T14:51:21Z Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth Wolanski, N. Harnack's inequality Local bounds Variable exponent spaces We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, Δ<inf>p(x)</inf>u := div(|∇<inf>u</inf>|p(x)-2∇<inf>u</inf>) = f(x) in Ω for which we prove Harnack's inequality when f ∈ Lq<inf>0</inf> (Ω) if max {1, N/p<inf>1</inf>} < q<inf>0</inf> ≤ ∞. The constant in Harnack's inequality depends on u only through |||u|p(x)||p<inf>2</inf>-p<inf>1</inf><inf>L1(Ω)</inf>. Dependence of the constant on u is known to be necessary in the case of variable p(x). As in previous papers, log-Hölder continuity on the exponent p(x) is assumed. We also prove that weak solutions are locally bounded and Hölder continuous when f ∈ Lq<inf>0</inf>(x)(Ω) with q<inf>0</inf> ∈ c(Ω) and max{1,N/p(x)} < q<inf>0</inf>(x) in Ω. These results are then generalized to elliptic equations div A(x,u,∇<inf>u</inf>) = B(x,u,∇<inf>u</inf>) with p(x)-type growth. Fil:Wolanski, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00416932_v56_n1_p73_Wolanski
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Harnack's inequality
Local bounds
Variable exponent spaces
spellingShingle Harnack's inequality
Local bounds
Variable exponent spaces
Wolanski, N.
Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth
topic_facet Harnack's inequality
Local bounds
Variable exponent spaces
description We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, Δ<inf>p(x)</inf>u := div(|∇<inf>u</inf>|p(x)-2∇<inf>u</inf>) = f(x) in Ω for which we prove Harnack's inequality when f ∈ Lq<inf>0</inf> (Ω) if max {1, N/p<inf>1</inf>} < q<inf>0</inf> ≤ ∞. The constant in Harnack's inequality depends on u only through |||u|p(x)||p<inf>2</inf>-p<inf>1</inf><inf>L1(Ω)</inf>. Dependence of the constant on u is known to be necessary in the case of variable p(x). As in previous papers, log-Hölder continuity on the exponent p(x) is assumed. We also prove that weak solutions are locally bounded and Hölder continuous when f ∈ Lq<inf>0</inf>(x)(Ω) with q<inf>0</inf> ∈ c(Ω) and max{1,N/p(x)} < q<inf>0</inf>(x) in Ω. These results are then generalized to elliptic equations div A(x,u,∇<inf>u</inf>) = B(x,u,∇<inf>u</inf>) with p(x)-type growth.
format JOUR
author Wolanski, N.
author_facet Wolanski, N.
author_sort Wolanski, N.
title Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth
title_short Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth
title_full Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth
title_fullStr Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth
title_full_unstemmed Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth
title_sort local bounds, harnack's inequality and hölder continuity for divergence type elliptic equations with non-standard growth
url http://hdl.handle.net/20.500.12110/paper_00416932_v56_n1_p73_Wolanski
work_keys_str_mv AT wolanskin localboundsharnacksinequalityandholdercontinuityfordivergencetypeellipticequationswithnonstandardgrowth
_version_ 1807320516310073344

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