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# Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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1864-8266
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Volume 10, Issue 1

# Continuity properties of solutions to the p-Laplace system

Angela Alberico
• Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Sezione di Napoli, Via P. Castellino 111, 80131 Napoli, Italy
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• Other articles by this author:
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/ Andrea Cianchi
• Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy
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/ Carlo Sbordone
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• Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy
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Published Online: 2015-11-17 | DOI: https://doi.org/10.1515/acv-2015-0029

## Abstract

A sharp integrability condition on the right-hand side of the p-Laplace system for all its solutions to be continuous is exhibited. Their uniform continuity is also analyzed and estimates for their modulus of continuity are provided. The relevant estimates are shown to be optimal as the right-hand side ranges in classes of rearrangement-invariant spaces, such as Lebesgue, Lorentz, Lorentz–Zygmund, and Marcinkiewicz spaces, as well as some customary Orlicz spaces.

MSC 2010: 35B65; 35J60; 46E30

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## About the article

Received: 2015-07-22

Accepted: 2015-10-12

Published Online: 2015-11-17

Published in Print: 2017-01-01

This research was partly supported by the research project of MIUR (Italian Ministry of Education, University and Research) PRIN 2012, no. 2012TC7588, “Elliptic and Parabolic Partial Differential Equations: Geometric Aspects, Related Inequalities, and Applications”, and by the GNAMPA (National Group for Mathematical Analysis, Probability and their Applications) of the Italian INdAM (National Institute of High Mathematics).

Citation Information: Advances in Calculus of Variations, Volume 10, Issue 1, Pages 1–24, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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