On the solvability of a two-dimensional Ventcel problem with variable coefficients

Abstract

This paper deals with the following mixed boundary value problem \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} -\Delta u = f &\mbox{in $\Omega$,} \\ u = \varphi &\mbox{on $\Gamma_{\! D}$,} \\ u_\nu - a_2 \, \Delta_{\tau \,} u + a_0 \, u = g &\mbox{on $\Gamma_{\! \nu}$,} \end{cases} \end{equation} where $\Omega$ is some bounded domain of $\mathbb{R}^2$ with $\partial \Omega=\Gamma_{\!D}\cup \Gamma_{\! \nu}$, $\nu$ indicating the normal unit vector to $\Gamma_{\! \nu}$ and $\Delta_\tau$ the Laplace--Beltrami operator along~$\Gamma_{\! \nu}$. Additionally, $f(\bf x)$, $\varphi(\bf x)$, $a_2(\bf x)$, $a_0(\bf x)$ and $g(\bf x)$ are convenient functions defined on $\Omega$, $\Gamma_{\!D}$ and $\Gamma_{\! \nu}$, and ${\bf x} = (x,y)$ denotes a two-dimensional array. Under suitable assumptions on the data, we first give the definition of a weak solution $u$ to the problem and then we prove that it is uniquely solvable. Further, we consider a particular case of \eqref{ProblemAbstract} arising in real-world applications: we discuss the resulting model and provide an explicit solution.