Rigidity results for the capillary overdetermined problem
Yuanyuan Lian, Pieralberto Sicbaldi
TL;DR
The paper studies the general capillary overdetermined problem with a mean curvature term depending on height, proving a sharp rigidity result in dimension two: under suitable boundedness and structural conditions on $f$ and its primitive $F$, a capillary graph defined on an unbounded, connected domain must be a half-plane with a parallel (one-dimensional) height profile. The authors develop a Modica-type $P$-function estimate to obtain a maximum-principle based rigidity, establish gradient bounds when $f'\le 0$, and perform a blow-up analysis to construct and characterize parallel solutions in a half-plane. A variational construction yields radial solutions in balls that approximate parallels, enabling a limiting argument to link the global geometry of $\Omega$ to the ODE governing parallel profiles. Consequently, the classical capillary problem (linear $f$) admits the same rigidity: the infinite plate must be planar if a capillary graph exists with nonzero contact angle, highlighting a deep connection between local curvature conditions and global domain geometry.
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 $\mathbb{R}$, $ν$ is the exterior unit normal, $κ$ is a constant and $Ω\subset \mathbb{R}^n$ is a $C^1$ domain. Our main theorem states that if $n=2, κ\neq 0$, $\partial Ω$ is unbounded and connected, $|\nabla u|$ is bounded and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\geq \left(1+κ^2\right)^{-\frac12} -1$, then $Ω$ 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 $f'(u) <0$. Moreover we study a Modica type estimate for the above overdetermined problem that allows us to prove that, unless $Ω$ is a half-space, the mean curvature of $\partial Ω$ is strictly negative under the assumption that $κ\neq 0$ and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\geq \left(1+κ^2\right)^{-\frac12} -1$. Our results have an interesting physical application to the classical capillary overdetermined problem, i.e., the case where $f$ is linear.