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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.17556 |
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| _version_ | 1866909591882694656 |
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| author | Bögelein, Verena Duzaar, Frank Treu, Giulia |
| author_facet | Bögelein, Verena Duzaar, Frank Treu, Giulia |
| contents | We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $$ \partial_tu-\text{div}_x \nabla_ξf(\nabla u)=0 $$ in a space-time cylinder $Ω_T=Ω\times (0,T)$, subject to time-dependent boundary data $g\colon \partial_{\mathcal{P}}Ω_T\to \mathbf{R}$ prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data $g$ along the lateral boundary $\partialΩ\times (0,T)$. More precisely, we require that for each fixed $t\in [0,T)$, the graph of $g(\cdot ,t)$ over $\partialΩ$ admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parabolic PDEs with Dynamic Data under a Bounded Slope Condition Bögelein, Verena Duzaar, Frank Treu, Giulia Analysis of PDEs 2010 35A01, 35K61, 35K86, 49J40 We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $$ \partial_tu-\text{div}_x \nabla_ξf(\nabla u)=0 $$ in a space-time cylinder $Ω_T=Ω\times (0,T)$, subject to time-dependent boundary data $g\colon \partial_{\mathcal{P}}Ω_T\to \mathbf{R}$ prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data $g$ along the lateral boundary $\partialΩ\times (0,T)$. More precisely, we require that for each fixed $t\in [0,T)$, the graph of $g(\cdot ,t)$ over $\partialΩ$ admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers. |
| title | Parabolic PDEs with Dynamic Data under a Bounded Slope Condition |
| topic | Analysis of PDEs 2010 35A01, 35K61, 35K86, 49J40 |
| url | https://arxiv.org/abs/2504.17556 |