Weighted energy-dissipation functionals for doubly nonlinear evolution

Abstract

This paper is concerned with the Weighted Energy-Dissipation (WED) functional approach to doubly nonlinear evolutionary problems. This approach consists in minimizing (WED) functionals defined over entire trajectories. We present the features of the WED variational formalism and analyze the related Euler-Lagrange problems. Moreover, we check that minimizers of the WED functionals converge to the corresponding limiting doubly nonlinear evolution. Finally, we present a discussion on the functional convergence of sequences of WED functionals and present some application of the abstract theory to nonlinear PDEs.

A variational principle for doubly nonlinear evolution

“…Finally, the abstract doubly nonlinear case (1 < p < ∞) is addressed in [7]. The present analysis does not follow from the results of [7] as the nonlinearities here are stronger than the ones which are allowed in [7] and hence require extra care. We shall mention that the existence for (1) has been already tackled in [8, Thm.…”

“…The rate-independent case (p = 1) has been considered by Mielke and Ortiz [5] and detailed in [6]. Finally, the abstract doubly nonlinear case (1 < p < ∞) is addressed in [7]. The present analysis does not follow from the results of [7] as the nonlinearities here are stronger than the ones which are allowed in [7] and hence require extra care.…”

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

“…The existence of a strong solution u to (3.1)-(3.3) in the sense of Definition 1 follows by a direct application of Proposition 6 (for checking that assumptions of Proposition 6 are satisfied we refer the reader to [2,3]). We recall that, if u solves (3.1)-(3.3) in the sense of Definition 1, then,…”

A variational view at the time-dependent minimal surface equation

“…This note concerns the L 2 Problem (1.1) admits, in general, no classical solution unless ∂ is (basically) of non-negative mean curvature (but see also [17,23,24] for more general conditions). Indeed, as the functional A is convex and proper but fails to be lower semicontinuous on L 2 ( ), its gradient flow generally does not admit strong solutions.…”

“…As mentioned, the general discussion of the WED functional approach to gradient flows is in [28] whereas two applications are in [14]. As for the doubly nonlinear dissipative evolution case, one shall mention the rate-independent theory of Mielke & Ortiz [26] (see also [27]), as well as the general convergence results of [1,2]. Finally, the semilinear hyperbolic case has been tackled via the WED approach in [30].…”