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.