Título: Rigidity results for the capillary overdetermined problem.
Conferenciante: Yuanyuan Lian (Universidad de Granada).
Resumen (Descargar resumen aquí):
In this paper we obtain rigidity results for bounded positive solutions of the general capillary overdetermined problem:
\begin{equation}
\begin{cases}
\text{div} \left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right) + f(u) = 0 &\text{ in } \Omega, \\
u = 0 &\text{ on } \partial\Omega, \\
\partial_{\nu} u = \kappa &\text{ on } \partial\Omega,
\end{cases}
\tag{1}
\end{equation}
where \( f \) is a given \( C^1 \) function in \( \mathbb{R} \), \( \nu \) is the exterior unit normal, \( \kappa \) is a constant and \( \Omega \subset \mathbb{R}^n \) is a \( C^1 \) domain. Our main theorem states that if \( n = 2 \), \( \kappa \neq 0 \), \( \partial\Omega \) is unbounded and connected, \( |\nabla u| \) is bounded, and there exists a nonpositive primitive \( F \) of \( f \) such that \(F(0) \geq (1 + \kappa^2)^{-\frac{1}{2}} - 1\), then \( \Omega \) must be a half-plane and \( u \) is a parallel solution. In other words, under our assumptions, if a capillary graph has the property that its mean curvature depends only on the height, then it is the graph of a one-dimensional function. We also prove the boundedness of the gradient of solutions of (1) when \( f'(u) 0 \). Moreover, we study a Modica type estimate for the overdetermined problem (1) that allows us to prove that, unless \( \Omega \) is a half-space, the mean curvature of \( \partial\Omega \) is strictly negative under the assumption that \( \kappa \neq 0 \) and there exists a nonpositive primitive \( F \) of \( f \) such that \( F(0) \geq (1 + \kappa^2)^{-\frac{1}{2}} - 1\). Our results have an interesting physical application to the classical capillary overdetermined problem, i.e., the case where \( f \) is linear.
Contact: yuanyuanlian@correo.ugr.es
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