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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

IMPACT FACTOR 2017: 1.676

CiteScore 2017: 1.30

SCImago Journal Rank (SJR) 2017: 2.045
Source Normalized Impact per Paper (SNIP) 2017: 1.138

Mathematical Citation Quotient (MCQ) 2017: 1.15

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Volume 9, Issue 1


A Neumann problem involving the p(x)-Laplacian with p = ∞ in a subdomain

Yiannis Karagiorgos
  • Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780, Greece
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Nikos Yannakakis
Published Online: 2014-12-11 | DOI: https://doi.org/10.1515/acv-2014-0003


In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p(x)-Laplacian is involved and p = ∞ in a subdomain. By considering a suitable sequence pk of bounded variable exponents such that pkp and replacing p with pk in the original problem, we prove the existence of a solution uk for each of those intermediate ones. We show that the limit of (uk) exists and after giving a variational characterization of it in the part of the domain where p is bounded, we show that it is a viscosity solution in the part where p = ∞. Finally, we formulate the problem of which this limit function is a solution in the viscosity sense.

Keywords: Neumann problem; variable exponent; p(x)-Laplacian; viscosity solution; infinity Laplacian; infinity harmonic function

MSC: 35J66; 35D40; 35D30

About the article

Received: 2014-02-24

Revised: 2014-10-03

Accepted: 2014-10-08

Published Online: 2014-12-11

Published in Print: 2016-01-01

Citation Information: Advances in Calculus of Variations, Volume 9, Issue 1, Pages 65–76, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2014-0003.

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