
Journal of Convex Analysis 25 (2018), No. 3, 861898 Copyright Heldermann Verlag 2018 EllipticRegularization of Nonpotential Perturbations of DoublyNonlinear Flows of Nonconvex Energies: A Variational Approach Goro Akagi Mathematical Institute, Tohoku University, Aoba, Sendai 9808578, Japan akagi@m.tohoku.ac.jp Stefano Melchionna Faculty of Mathematics, University of Vienna, OskarMorgensternPlatz 1, 1090 Wien, Austria stefano.melchionna@univie.ac.at [Abstractpdf] This paper presents a variational approach to doublynonlinear (gradient) flows (P) of nonconvex energies along with nonpotential perturbations (i.e., perturbation terms without any potential structures). An ellipticintime regularization of the original equation ${\rm (P)}_\varepsilon$ is introduced, and then, a variational approach and a fixedpoint argument are employed to prove existence of strong solutions to ${\rm (P)}_\varepsilon$. More precisely, we introduce a family of functionals (defined over entire trajectories) parametrized by a small parameter $\varepsilon$, whose EulerLagrange equation corresponds to the ellipticintime regularization of an unperturbed (i.e.~without nonpotential perturbations) doublynonlinear flow. Secondly, due to the presence of nonpotential perturbation, a fixedpoint argument is performed to construct strong solutions $u_\varepsilon$ to the ellipticintime regularized equations ${\rm (P)}_\varepsilon$. Finally, a strong solution to the original equation (P) is obtained by passing to the limit of $u_\varepsilon$ as $\varepsilon\to 0$. Applications of the abstract theory developed in the present paper to concrete PDEs are also exhibited. [ Fulltextpdf (228 KB)] for subscribers only. 