The obstacle problem for singular doubly nonlinear equations of porous medium type

Abstract

In this paper we prove the existence of variational solutions to the obstacle problem associated with doubly nonlinear equations \partial_t (|u|^{m-1}u) - \mathrm {div}(D_\xi f(Du)) = 0 with m > 1 and a convex function f satisfying a standard p -growth condition for an exponent p \in (1,\infty) in a bounded space-time cylinder \Omega_T := \Omega \times (0,T) . The obstacle function \psi and the boundary values g are time dependent. The proof relies on a nonlinear version of the method of minimizing movements.

The obstacle problem for degenerate doubly nonlinear equations of porous medium type

“…By the nonlinear minimizing movements scheme developed in [11] and suitable approximation arguments, the existence of signed or vector-valued variational solutions to the Cauchy-Dirichlet problem with time-dependent boundary values and the existence of solutions to the obstacle problem with time-dependent obstacle function have been established, cf. [26,27]. More precisely, the boundary values and the obstacle function are contained in the space L p (0, T;W 1,p (Ω)) with time derivative in L 1 (0, T;L m+1 (Ω)) and initial values in L m+1 (Ω) .…”

Functional inequalities and applications to doubly nonlinear diffusion equations

“…What is more, then evolution equation (1) requires different methods for 1 < p < 2 and 2 < p being called singular and degenerate, respectively. Studies on solutions to equations like (1) attract deep attention of various groups developing their theory from different points of view [2,4,14,39,40,41,49,54,60,67]. The issue of convergence of solutions to a self-similar profile to nonlinear diffusion equations has been studied e.g.…”

Functional inequalities and applications to doubly nonlinear diffusion equations

“…What is more, then evolution equation (1) requires different methods for 1 < p < 2 and 2 < p being called singular and degenerate, respectively. Studies on solutions to equations like (1) attract deep attention of various groups developing their theory from different points of view [2,4,14,39,40,41,48,53,59,66]. The issue of convergence of solutions to a self-similar profile to nonlinear diffusion equations has been studied e.g.…”