An abstract transmission problem in a thin layer, I: Sharp estimates
The authors study the following abstract elliptic transmission problem \[ \begin{cases} u''_{-}(x)+Au_{-}(x)=-g_{-}(x),\quad x\in (-1,0),\\ u''_{+}(x)+Au_{+}(x)=-g_{+}(x),\quad x\in (0,\delta),\\ u_{-}(-1)=f_{-},\\ u'_{+}(\delta)=f_{+},\\ u_{-}(0)=u_{+}(0),\\ p_{-}u'_{-}(0)=p_{+}u'_{+}(0)+\chi, \end{cases} \] where \(A\) is a linear closed operator in a Banach space \(X\), \(p_{-},p_{+}\in\mathbb R^+\), \(\delta\) is a small parameter, \(g_{-}\in L^p((-1,0);X)\), \(g_{+}\in L^p((0,\delta);X)\), \(f_{+},f_{-},\chi\in X\). Existence, uniqueness, and maximal regularity of the solution \((u_{-},u_{+})\) of the above problem in the sense of \(L^p\)-spaces and its behavior for small values of the parameter \(\delta\) are established in the paper. The main tools of the authors' approach are impedance and admittance operators (i.e., Dirichlet-to-Neumann and Neumann-to-Dirichlet operators) and \(H^{\infty}\) functional calculus for sectorial operators in Banach spaces. First, a few physical models are presented in the paper as a motivation of the abstract study. At the end of the paper, the authors solve these model problems using the obtained abstract results. The authors also announced in the paper, that in a forthcoming paper they will study the limit problem for \(\delta\to 0\).
scientific article; zbMATH DE number 5949072
scientific article