Year: 2020
Type: preprint
Abstract: This paper deals with the following mixed boundary value problem \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} -Δu = f &\mbox{in $Ω$,} \\ u = φ&\mbox{on $Γ_{\! D}$,} \\ u_ν- a_2 \, Δ_{τ\,} u + a_0 \, u = g &\mbox{on $Γ_{\! ν}$,} \end{cases} \end{equation} where $Ω$ is some bounded domain of $\mathbb{R}^2$ with $\partial Ω=Γ_{\!D}\cup Γ_{\! ν}$, $ν$ indicating the normal unit vector to $Γ_{\! ν}$ and $Δ_τ$ the Laplace--Beltrami operator along~$Γ_{\! ν}$. Additionally, $f(\bf x)$, $φ(\bf x)$, $a_2(\bf x)$, $a_0(\bf x)$ and $g(\bf x)$ are convenient functions defined on $Ω$, $Γ_{\!D}$ and $Γ_{\! ν}$, 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.
Source: arXiv (Cornell University)
Authors: Greco, Antonio, Viglialoro, Giuseppe
Institution: University of Cagliari
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3005601534