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# Analysis

### International mathematical journal of analysis and its applications

CiteScore 2018: 0.72

SCImago Journal Rank (SJR) 2018: 0.363
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Volume 37, Issue 4

# Existence of variational solutions for time dependent integrands via minimizing movements

Leah Schätzler
• Corresponding author
• Department Mathematik, Friedrich-Alexander-Universität Erlangen–Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany
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Published Online: 2017-10-25 | DOI: https://doi.org/10.1515/anly-2017-0047

## Abstract

We prove the existence of variational solutions to equations of the form

${\partial }_{t}u-\mathrm{div}\left({D}_{\xi }f\left(x,t,Du\right)\right)=0,$

where the function f merely satisfies a p-growth condition and is convex with respect to the gradient variable. In particular, we do not require any regularity assumption with respect to time. We obtain an existence result for integrands that are Lipschitz continuous in time via the method of minimizing movements. For the general existence result, we show stability of solutions with respect to approximation of the integrands. In this context, we prove a result related to Γ-convergence that is also valid for functionals with $\left(p,q\right)$-growth.

Keywords: minimizing movements

MSC 2010: 35K20; 49J40

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Accepted: 2017-09-21

Published Online: 2017-10-25

Published in Print: 2017-11-01

Citation Information: Analysis, Volume 37, Issue 4, Pages 199–222, ISSN (Online) 2196-6753, ISSN (Print) 0174-4747,

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© 2017 Walter de Gruyter GmbH, Berlin/Boston.