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, Noemi Irene
Publicado: 2015
Materias:
Harnack's inequality
Local bounds
Variable exponent spaces
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00416932_v56_n1_p73_Wolanski
http://hdl.handle.net/20.500.12110/paper_00416932_v56_n1_p73_Wolanski
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00416932_v56_n1_p73_Wolanski
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spelling paper:paper_00416932_v56_n1_p73_Wolanski2023-06-08T15:04:48Z Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth Wolanski, Noemi Irene 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. 2015 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00416932_v56_n1_p73_Wolanski 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, Noemi Irene
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.
author Wolanski, Noemi Irene
author_facet Wolanski, Noemi Irene
author_sort Wolanski, Noemi Irene
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
publishDate 2015
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00416932_v56_n1_p73_Wolanski
http://hdl.handle.net/20.500.12110/paper_00416932_v56_n1_p73_Wolanski
work_keys_str_mv AT wolanskinoemiirene localboundsharnacksinequalityandholdercontinuityfordivergencetypeellipticequationswithnonstandardgrowth
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