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

This paper deals with the following mixed boundary value problem \begin{equation}\label{ProblemAbstract} \tag{$\mathrm{◊}$} \begin{cases} -\Delta u = f &\mbox{in $\mathrm{\Omega }$,} \ u = \varphi &\mbox{on ${\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}D}$,} \ u_\nu

• a_2 , \Delta_{\tau ,} u + a_0 , u = g &\mbox{on ${\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}\nu }$,} \end{cases} \end{equation} where $\mathrm{\Omega }$ is some bounded domain of ${\mathbb{R}}^2$ with $\mathrm{\partial }\mathrm{\Omega }={\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}D}\cup {\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}\nu }$, $\nu$ indicating the normal unit vector to ${\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}\nu }$ and ${\mathrm{\Delta }}_{\tau }$ the Laplace–Beltrami operator along~${\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}\nu }$. Additionally, $f(\mathbf{x})$, $\phi (\mathbf{x})$, $a_2(\mathbf{x})$, $a_0(\mathbf{x})$ and $g(\mathbf{x})$ are convenient functions defined on $\mathrm{\Omega }$, ${\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}D}$ and ${\mathrm{\Gamma }}_{\text{\hspace{0.17em}⁣}\nu }$, 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.