A variational approach to doubly nonlinear equations
type: Item
instance of: scholarly article
title: A variational approach to doubly nonlinear equations
zbMATH Open document ID: 1407.35123
DOI: 10.4171/RLM/832
author: Verena Bögelein
author: Frank Duzaar
author: Paolo Marcellini
author: Christoph Scheven
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2019-03-05T00:00:00Z
review text: Summary: This article presents a variational approach to the existence of solutions to equations of Porous Medium type. More generally, the method applies also to doubly nonlinear equations with a nonlinearity in \(u\) and \(Du\), whose prototype is given by \[ \partial_t u^m-\mathrm {div}(|D u|^{p-2}D u) = 0, \] where \(m 0\) and \(p 1\). The technique relies on a nonlinear version of the Minimizing Movement Method which has been introduced in [the authors, Arch. Ration. Mech. Anal. 229, No. 2, 503--545 (2018; Zbl 1394.35349)] in the context of doubly nonlinear equations with general nonlinearities \(\partial_t b(u)\) and more general operators with variational structure. The aim of this article is twofold. On the one hand it provides an introduction to variational solutions and outlines the method developed in [loc. cit.]. In addition, we extend the results of [loc. cit.] to initial data with potentially infinite energy. This requires a detailed discussion of the growth conditions of the variational energy integrand. The approach is flexible enough to treat various more general evolutionary problems, such as signed solutions, obstacle problems, time dependent boundary data or problems with linear growth.
Mathematics Subject Classification ID: 35K86
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 7032422
zbMATH Keywords: nodal solutions
zbMATH Keywords: indefinite potential
zbMATH Keywords: nonhomogeneous differential operator
zbMATH Keywords: nonlinear regularity theory
zbMATH Keywords: truncation and cut-off techniques
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/rlm/832
OpenAlex ID: W2894717703
Wikidata QID: Q128667532
description: scientific article; zbMATH DE number 7032422