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Differentiability properties of solutions of the equation \(-\epsilon ^ 2\Delta u+ru=f(x,y)\) in a square. (English) Zbl 0732.35020

For the solution of a singularly perturbed elliptic boundary value problem \(-\epsilon^ 2\Delta u+r(x,y)u=f\) with nonhomogeneous Dirichlet data on the unit square (0,1)\(\times (0,1)\) in \({\mathbb{R}}^ 2\), Butuzov gave an asymptotic expansion in terms of \(\epsilon\) which includes usual boundary layer functions and “corner layer” functions. In this article it is shown that when \(r\equiv const\). the expansion can be termwise differentiated. This establishes the singular perturbation effect of the corner singularities of the Dirichlet problem on a domain with rectangular corners.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B25 Singular perturbations in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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