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{$◊$} \begin{cases} -Δu = f &\mbox{in $\Omega$,} \\ u = φ&\mbox{on ${\Gamma }_D$,} \\ u_ν- a_2 \, Δ_{τ\,} 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, $f(\mathbf{x})$, $\phi (\mathbf{x})$, $a_2(\mathbf{x})$, $a_0(\mathbf{x})$ and $g(\mathbf{x})$ are convenient functions defined on $\Omega$, ${\Gamma }_D$ and ${\Gamma }_{\nu }$, and $\mathbf{x}=(x,y)$ 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.