Title: On the Solution of Elliptic Partial Differential Equations on Regions with Corners III: Curved Boundaries
Author(s): Serkh, Kirill
In this report we investigate the solution of boundary value problems for elliptic partial differential equations on domains with corners. Previously, we observed that when, in the case of polygonal domains, the boundary value problems are formulated as boundary integral equations of classical potential theory, the solutions are representable by series of certain elementary functions. Here, we extend this observation to the general case of regions with boundaries consisting of analytic curves meeting at corners. We show that the solutions near the corners have the same leading terms as in the polygonal case, plus a series of corrections involving products of the leading terms with integer powers and powers of logarithms. Furthermore, we show that if the curve in the vicinity of a corner approximates a polygon to order \(k\), then the correction added to the leading terms will vanish like \(O(t^k)\), where \(t\) is the distance from the corner.