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# Advances in Calculus of Variations

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Volume 11, Issue 1

# Relative ∞-capacity and its affinization

Renjin Jiang
• School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Beijing 100875, P. R. China
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/ Jie Xiao
• Corresponding author
• Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada
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/ Dachun Yang
• School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Beijing 100875, P. R. China
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Published Online: 2016-09-17 | DOI: https://doi.org/10.1515/acv-2016-0023

## Abstract

In this paper, the so-called relative $\mathrm{\infty }$-capacity is introduced and investigated in a close connection to the viscosity solution of the $\mathrm{\infty }$-Laplace equation. We not only show that the relative $\mathrm{\infty }$-capacity equals the limit of the p-th root of the relative p-capacity as $p\to \mathrm{\infty }$ and hence has a simple geometric characterization in terms of the Euclidean distance, but also establish several basic properties for the relative $\mathrm{\infty }$-capacity. Consequently, we apply the relative $\mathrm{\infty }$-capacity to the embedding theory of the $\mathrm{\infty }$-Sobolev space. More geometrically, we affinize the relative $\mathrm{\infty }$-capacity and its fundamental features as much as possible.

MSC 2010: 53A15; 47A75; 35J70

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## About the article

Received: 2016-05-13

Revised: 2016-08-09

Accepted: 2016-08-11

Published Online: 2016-09-17

Published in Print: 2018-01-01

Funding Source: Natural Sciences and Engineering Research Council of Canada

Award identifier / Grant number: FOAPAL 202979463102000

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11301029

Award identifier / Grant number: 11171027

Award identifier / Grant number: 11361020

J. Xiao was supported by NSERC of Canada (FOAPAL No. 202979463102000); R. Jiang and D. Yang were partially supported by the National Natural Science Foundation of China (Nos. 11301029, 11171027 and 11361020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120003110003) and the Fundamental Research Funds for Central Universities of China (Nos. 2012LYB26 and 2013YB60).

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 1, Pages 95–110, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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