论文标题
关于具有可变系数的二维通风仪问题的可溶性
On the solvability of a two-dimensional Ventcel problem with variable coefficients
论文作者
论文摘要
本文处理以下混合边界价值问题\ begin {equination} \ label {QuasicaBstract} \ tag {$ \ diamond $} \ begin {cases}-ΔU= f＆\ mbox {in $ = = up =φ＆\ mbox {in $ =φ＆\ mbox {on $γ_ { d} $，} \\u_ν-a_2 \，δ_{τ\，} u + a_0 \，u = g＆\ mbox {on $γ_{\！ ν} $，} \ end {cases} \ end {qore}，其中$ω$是$ \ mathbb {r}^2 $的某些有限域，带有$ \ partialω=γ_=γ_{\！d} \！ ν} $，$ν$表示普通单位向量为$γ_ {\！ ν} $和$Δ_τ$沿〜$γ_ {\！ ν} $。此外，$ f（\ bf x）$，$φ（\ bf x）$，$ a_2（\ bf x）$，$ a_0（\ bf x）$和$ g（\ bf x）$是$ω$，$γ_{\！d} $和$γ_ _ {\！ ν} $，$ {\ bf x} =（x，y）$表示二维数组。在数据上的合适假设下，我们首先将弱解决方案$ u $的定义用于问题，然后证明它是唯一可解决的。此外，我们考虑了在现实世界应用程序中引起的\ eqref {QuessionAbstract}的特定情况：我们讨论结果模型并提供明确的解决方案。
This paper deals with the following mixed boundary value problem \begin{equation}\label{ProblemAbstract} \tag{$\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.
