A variational approach to doubly nonlinear equations. (English) Zbl 1407.35123
Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 29, No. 4, 739-772 (2018).
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.
MSC:
| 35K86 | Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators |
| 49J40 | Variational inequalities |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
Keywords:
nodal solutions; indefinite potential; nonhomogeneous differential operator; nonlinear regularity theory; truncation and cut-off techniquesCitations:
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