Existence of variational solutions to doubly nonlinear systems in nondecreasing domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912876113952768 |
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| author | Schätzler, Leah Scheven, Christoph Siltakoski, Jarkko Stanko, Calvin |
| author_facet | Schätzler, Leah Scheven, Christoph Siltakoski, Jarkko Stanko, Calvin |
| contents | 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_ξf(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 $x \mapsto f(x,u,ξ)$ is integrable, that $(u,ξ) \mapsto f(x,u,ξ)$ 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 $V^{p,0}(E) \subset L^p(0,T;W^{1,p}(Ω,\mathbb{R}^N))$ that encodes zero boundary values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of variational solutions to doubly nonlinear systems in nondecreasing domains Schätzler, Leah Scheven, Christoph Siltakoski, Jarkko Stanko, Calvin Analysis of PDEs 35A01, 35A15, 35K51, 35K55, 49J40 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_ξf(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 $x \mapsto f(x,u,ξ)$ is integrable, that $(u,ξ) \mapsto f(x,u,ξ)$ 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 $V^{p,0}(E) \subset L^p(0,T;W^{1,p}(Ω,\mathbb{R}^N))$ that encodes zero boundary values. |
| title | Existence of variational solutions to doubly nonlinear systems in nondecreasing domains |
| topic | Analysis of PDEs 35A01, 35A15, 35K51, 35K55, 49J40 |
| url | https://arxiv.org/abs/2505.00148 |