On boundary value problem with singular inhomogeneity concentrated on the boundary
The paper deals with boundary homogenization for a linear elliptic equation. The authors consider the Poisson equation NEWLINE\[NEWLINE-\Delta u+ (\varepsilon\delta)^{-\chi}\rho_{\varepsilon, \delta} u= f_{\varepsilon, \delta}\tag{1}NEWLINE\]NEWLINE in an open domain \(\Omega\) of \(\mathbb{R}^n\), \(n\geq 3\), with a piecewise smooth boundary. It is assumed that \(\Omega\) lies in the upper half space \(\mathbb{R}^{n-1}\times (0,+\infty)\) and NEWLINE\[NEWLINE\Gamma= \partial\Omega\cap (\mathbb{R}^{n+1}\times \{0\})= [-1/2, 1/2]^{n-1}\times \{0\}.NEWLINE\]NEWLINE Moreover \((\varepsilon\delta)^{-\chi}\rho_{\varepsilon, \delta}\) (with \(\|\rho_{\varepsilon, \delta}\|_{L^\infty}\leq C\)) is an asymptotically singular nonnegative density, periodic or locally periodic with respect to the \(n-1\) first variables, supported in a union of balls of radius \(c_0\delta\varepsilon\), centered at the points of \(\varepsilon\mathbb{Z}^n\cap F\); \(f_{\varepsilon,\delta}- f\) enjoys a localization and periodicity property of the same kind for a fixed \(f\in L^2(\Omega)\).NEWLINENEWLINE Calling \(u_{\varepsilon,\delta}\) the solution of (1) submitted to some boundary condition that alternates rapidly (with periods in \(\varepsilon\mathbb{Z}^n\)) between Dirichlet and Neumann on \(\Gamma\), the authors are interested in the behavior of \(u_{\varepsilon,\delta}\) as both \(\delta\) and \(\varepsilon\) tend to \(0\). Assuming that \(\delta^{n-2}/\varepsilon\) tends to \(k\in [0,+\infty]\), they prove that \(u_{\varepsilon,\delta}\) weakly converges in \(H^1\) topology to the solution of a limit problem whose boundary condition depends on \(k\) and \(\chi\).NEWLINENEWLINE The authors introduce a boundary layer unfolding operator and apply an unfolding procedure to prove their convergence results. Strong convergence is also considered.
scientific article; zbMATH DE number 6064164
scientific article