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

Abstract

AbstractWe prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation $$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}(Df(Du)) = 0, \end{aligned}$$ ∂ t b ( u ) - div ( D f ( D u ) ) = 0 , where the nonlinearity $$b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}$$ b : R ≥ 0 → R ≥ 0 is increasing, piecewise $$C^1$$ C 1 and satisfies a polynomial growth condition. The prototype is $$b(u) := u^m$$ b ( u ) : = u m with $$m \in (0,1)$$ m ∈ ( 0 , 1 ) . Further, $$f :\mathbb {R}^n \rightarrow \mathbb {R}_{\ge 0}$$ f : R n → R ≥ 0 is convex and fulfills a standard p-growth condition. The proof relies on a nonlinear version of the method of minimizing movements.