×

Asymptotics of the first boundary problem for elliptic equations in a domain with thin covering. (English) Zbl 0699.35059

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\) and let \(\Omega_{\varepsilon}\) be its thin covering of thickness \(\varepsilon\phi(x)\). The author investigates the equation \((A_{ij}(x)u^{\varepsilon}_{,j})_{,i} = f(x)\) in \(\Omega \cup \Omega_{\varepsilon}\), where \(A_{ij}(x)=a_{ij}(x)\) in \(\Omega\) and \(A_{ij}(x) = \varepsilon b_{ij}(x)\) in \(\Omega_{\epsilon}\), with the boundary condition \(u^{\epsilon}(x)=T(x)\), \(x\in \partial \Omega_{\epsilon}\) and interface conditions \(u^{\epsilon}\) and \(A_{ij}(x)u^{\epsilon}_{,j}n_ i\) continuous on \(\partial \Omega\). He shows that for \(\epsilon\) \(\to 0\) the family of solutions \(u^{\varepsilon}\) converges weakly in \(H^ 1(\Omega)\) to the solution of the problem \[ (a_{ij} u^0_{,j})_{,i} = f(x) \text{ in } \Omega,\quad a_{ij}u^ 0_{,j}n_ i = b_{ij}n_ in_ j(T(x) - u^ 0)/\phi (x) \text{ on } \partial \Omega. \]
Reviewer: O.Vejvoda

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
PDFBibTeX XMLCite
Full Text: EuDML