Calculus of Variations and Geometric Measure Theory

G. Akagi - V. Bögelein - A. Marveggio - U. Stefanelli

Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations

created by marveggio on 18 Jan 2024



Inserted: 18 jan 2024

Year: 2024

ArXiv: 2401.08856 PDF


We discuss a variational approach to doubly nonlinear wave equations of the form $\rho u_{tt} + g (u_t) - \Delta u + f (u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to $0$. The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit. Eventually, we investigate the viscous limit $\rho \to 0$, both at the functional level and on that of the equation.