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Global Hölder continuity of solutions to quasilinear equations with Morrey data

Department of Mathematical Sciences, and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea

E-mail Address: byun@snu.ac.kr

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Dian K. Palagachev

Politecnico di Bari, Dipartimento di Meccanica, Matematica e Management, Via Edoardo Orabona 4, 70125 Bari, Italy

E-mail Address: dian.palagachev@poliba.it

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, and

Pilsoo Shin

Department of Mathematics, Kyonggi University, Suwon 16227, Republic of Korea

E-mail Address: shinpilsoo.math@kgu.ac.kr

corresponding author.

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https://doi.org/10.1142/S0219199721500620Cited by:2 (Source: Crossref)

Abstract

We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$ -Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property.

We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^{p}$ -result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.

Keywords:

AMSC: 35J60, 35B65, 35R05, 35R06, 35B45, 35J92, 46E30

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