JournalsrlmVol. 29, No. 4pp. 739–772

A variational approach to doubly nonlinear equations

  • Verena Bögelein

    Universität Salzburg, Austria
  • Frank Duzaar

    Universität Erlangen-Nürnberg, Germany
  • Paolo Marcellini

    Università di Firenze, Italy
  • Christoph Scheven

    Universität Duisburg-Essen, Germany
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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 uu and DuDu, whose prototype is given by

tumdiv(Dup2Du)=0,\partial_t u^m-\mathrm {div}(|D u|^{p-2}D u) = 0,

where m>0m > 0 and p>1p > 1. The technique relies on a nonlinear version of the Minimizing Movement Method which has been introduced in [14] in the context of doubly nonlinear equations with general nonlinearities tb(u)\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 [14]. In addition, we extend the results of [14] 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.

Cite this article

Verena Bögelein, Frank Duzaar, Paolo Marcellini, Christoph Scheven, A variational approach to doubly nonlinear equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 4, pp. 739–772

DOI 10.4171/RLM/832