标题： 非减区域上双重非线性系统变分解的存在性 显示英文标题

标题： Existence of variational solutions to doubly nonlinear systems in nondecreasing domains

摘要： 对于$q \in (0, \infty)$，我们考虑在有界非柱形区域$E \subset \mathbb{R}^{n+1}$中形式为\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*}的双重非线性系统的柯西-狄利克雷问题。 我们假设$x \mapsto f(x,u,\xi)$是可积的，$(u,\xi) \mapsto f(x,u,\xi)$是凸的，并且$f$满足某个$p \in (1,\infty)$的$p$-强制性条件。然而，我们不对$f$施加上界的任何特定增长率条件。 对于仅满足$\mathcal{L}^{n+1}(\partial E) = 0$的非减区域，我们通过最小化运动方法的非线性版本证明了变分解$u \in C^{0}([0,T];L^{q+1}(E,\mathbb{R}^N))$的存在性。 此外，在关于$E$的额外假设以及$f$对$p$的增长条件下，我们证明了$|u|^{q-1}u$在编码零边界值的子空间$V^{p,0}(E) \subset L^p(0,T;W^{1,p}(\Omega,\mathbb{R}^N))$的对偶空间$(V^{p,0}(E))^{\prime}$中存在弱时间导数。 显示英文摘要

摘要： For $q \in (0, \infty)$, 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*} in a bounded noncylindrical domain $E \subset \mathbb{R}^{n+1}$. We assume that $x \mapsto f(x,u,\xi)$ is integrable, that $(u,\xi) \mapsto f(x,u,\xi)$ is convex, and that $f$ satisfies a $p$-coercivity condition for some $p \in (1,\infty)$. However, we do not impose any specific growth condition from above on $f$. For nondecreasing domains that merely satisfy $\mathcal{L}^{n+1}(\partial E) = 0$, we prove the existence of variational solutions $u \in C^{0}([0,T];L^{q+1}(E,\mathbb{R}^N))$ via a nonlinear version of the method of minimizing movements. Moreover, under additional assumptions on $E$ and a $p$-growth condition on $f$, we show that $|u|^{q-1}u$ admits a weak time derivative in the dual $(V^{p,0}(E))^{\prime}$ of the subspace $V^{p,0}(E) \subset L^p(0,T;W^{1,p}(\Omega,\mathbb{R}^N))$ that encodes zero boundary values.