Existence of variational solutions to doubly nonlinear systems in general noncylindrical domains

TL;DR

This paper proves the existence of variational solutions for doubly nonlinear systems in complex, noncylindrical domains, extending the theory to more general geometries and nonlinearities with specific growth conditions.

Contribution

It establishes the existence of variational solutions in noncylindrical domains under minimal boundary regularity and growth assumptions, including cases where the domain boundary has measure zero.

Findings

Existence of variational solutions in general noncylindrical domains.

Solutions have a distributional time derivative if the domain does not shrink too fast.

Under certain conditions, solutions are continuous in time.

Abstract

We consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_\xi f(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} with $q \in (0, \infty)$ in a bounded noncylindrical domain $E \subset \mathbb{R}^{n+1}$. Further, we suppose that $x \mapsto f(x,u,\xi)$ is integrable, that $(u,\xi) \mapsto f(x,u,\xi)$ is convex, and that $f$ satisfies a $p$-growth and -coercivity condition for some $p>\max \big\{ 1,\frac{n(q+1)}{n+q+1} \big\}$. Merely assuming that $\mathcal{L}^{n+1}(\partial E) = 0$, we prove the existence of variational solutions $u \in L^\infty\big( 0,T;L^{q+1}(E,\mathbb{R}^N)\big)$. If $E$ does not shrink too fast, we show that for the solution $u$ constructed in the first step, $\vert u \vert^{q-1}u$ admits a distributional time derivative. Moreover, under suitable conditions on $E$ and the stricter lower bound $p \geq \frac{(n+1)(q+1)}{n+q+1}$, $u$ is continuous with respect to time.