Rigidity results for the capillary overdetermined problem

Abstract

In this paper we obtain rigidity results for bounded positive solutions of the general capillary overdetermined problem \begin{equation} \left\{ \begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0 & \mbox{in }\; Ω,\\[1mm] u= 0 & \mbox{on }\; \partial Ω,\\[1mm] \partial_ν u=κ&\mbox{on }\; \partial Ω, \end{array}\right. \end{equation} where $f$ is a given $C^1$ function in $\mathbb{R}$, $\hat{I}½$ is the exterior unit normal, $\hat{I}º$ is a constant and $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ is a $C^1$ domain. Our main theorem states that if $n=2,\hat{I}º\ne 0$, $\partial \hat{I}\text{c◯}$ is unbounded and connected, $|\nabla u|$ is bounded and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\ge {\left(1+\hat{I}{º}^2\right)}^{-\frac{1}{2}}-1$, then $\hat{I}\text{c◯}$ 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 the above problem when $f^{\prime }(u)<0$. Moreover we study a Modica type estimate for the above overdetermined problem that allows us to prove that, unless $\hat{I}\text{c◯}$ is a half-space, the mean curvature of $\partial \hat{I}\text{c◯}$ is strictly negative under the assumption that $\hat{I}º\ne 0$ and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\ge {\left(1+\hat{I}{º}^2\right)}^{-\frac{1}{2}}-1$. Our results have an interesting physical application to the classical capillary overdetermined problem, i.e., the case where $f$ is linear.

42 pages