Convexity for a parabolic fully nonlinear free boundary problem with singular term
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909093587845120 |
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| author | Jeon, Seongmin Shahgholian, Henrik |
| author_facet | Jeon, Seongmin Shahgholian, Henrik |
| contents | In this paper, we study a parabolic free boundary problem in an exterior domain $$\begin{cases} F(D^2u)-\partial_tu=u^aχ_{\{u>0\}}&\text{in }(\mathbb R^n\setminus K)\times(0,\infty),\\ u=u_0&\text{on }\{t=0\},\\ |\nabla u|=u=0&\text{on }\partialΩ\cap(\mathbb R^n\times(0,\infty)),\\ u=1&\text{in }K\times[0,\infty).\end{cases}$$ Here, $a$ belongs to the interval $(-1,0)$, $K$ is a (given) convex compact set in $\mathbb R^n$, $Ω=\{u>0\}\supset K\times(0,\infty)$ is an unknown set, and $F$ denotes a fully nonlinear operator. Assuming a suitable condition on the initial value $u_0$, we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_02991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convexity for a parabolic fully nonlinear free boundary problem with singular term Jeon, Seongmin Shahgholian, Henrik Analysis of PDEs 35R35, 52A01 In this paper, we study a parabolic free boundary problem in an exterior domain $$\begin{cases} F(D^2u)-\partial_tu=u^aχ_{\{u>0\}}&\text{in }(\mathbb R^n\setminus K)\times(0,\infty),\\ u=u_0&\text{on }\{t=0\},\\ |\nabla u|=u=0&\text{on }\partialΩ\cap(\mathbb R^n\times(0,\infty)),\\ u=1&\text{in }K\times[0,\infty).\end{cases}$$ Here, $a$ belongs to the interval $(-1,0)$, $K$ is a (given) convex compact set in $\mathbb R^n$, $Ω=\{u>0\}\supset K\times(0,\infty)$ is an unknown set, and $F$ denotes a fully nonlinear operator. Assuming a suitable condition on the initial value $u_0$, we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time. |
| title | Convexity for a parabolic fully nonlinear free boundary problem with singular term |
| topic | Analysis of PDEs 35R35, 52A01 |
| url | https://arxiv.org/abs/2402.02991 |