A variational approach to symmetry, monotonicity, and comparison for doubly-nonlinear equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:456

Article

Melchionna, Stefano¹ 

[1] Univ Vienna, Fac Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

关键词: Qualitative properties;    Comparison principles;    Variational approach;    WED functionals;   

DOI  :  10.1016/j.jmaa.2017.07.052

来源: Elsevier

【 摘 要 】

We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter goes to zero. Qualitative properties and comparison principles can be easily proved for minimizers of the WED functional and, by passing to the limit, for the limiting problem. Several applications of the abstract results to systems of nonlinear PDEs and to fractional/nonlocal problems are presented. Eventually, we present some extensions of this approach in order to deal with rate independent systems and hyperbolic problems. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free