Bifurcation of overdetermined capillary problems in a strip domain

In this paper, we consider the classical overdetermined capillary problem: \begin{equation*} \begin{cases} $\mathrm{d}\mathrm{i}\mathrm{v}$ \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) - bu =0 &\mbox{in} Ω, \partial_ν u=κ&\mbox{on}\partialΩ, u=c &\mbox{on}\partialΩ, \end{cases} \end{equation*} where $b$, $c$ and $\kappa$ are positive constants, and $\mathrm{\Omega }\subset {\mathbb{R}}^2$. When $\mathrm{\Omega }$ is an infinite strip, i.e., a domain bounded by two parallel straight lines, there exists a unique one-dimensional solution (called the trivial solution) to this problem. By means of a bifurcation argument, we establish the existence of a critical period $T_{\ast }$ at which a branch of non-trivial solutions bifurcates from the trivial one. These solutions are genuinely two-dimensional and are defined in unbounded periodic domains $\mathrm{\Omega }$ that are diffeomorphic to an infinite strip, yet whose boundaries are no longer straight lines. This result offers a significant physical interpretation in the context of capillary phenomena.