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- W3005601534 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." @default.
- W3005601534 created "2020-02-24" @default.
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- W3005601534 date "2020-02-14" @default.
- W3005601534 modified "2023-09-27" @default.
- W3005601534 title "On the solvability of a two-dimensional Ventcel problem with variable coefficients" @default.
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