Abstract:In this paper, we consider the classical overdetermined capillary problem:
\begin{equation*}
\begin{cases}
\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) - bu =0 &~~\mbox{in}~~ \Omega,
\partial_{\nu} u=\kappa &~~\mbox{on}~~\partial\Omega,
u=c &~~\mbox{on}~~\partial\Omega,
\end{cases} \end{equation*}
where $b$, $c$ and $\kappa$ are positive constants, and $\Omega\subset \mathbb{R}^2$. When $\Omega$ 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 $T_*$ 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 $\Omega$ 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.
From: Yuanyuan Lian Dr. [view email]
[v1]
Thu, 18 Jun 2026 07:44:21 UTC (23 KB)
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