Title: On the solution of elliptic PDEs on three-dimensional regions with corners and conical points

Abstract: The solution of partial differential equations (PDEs) to high precision on regions with perfectly sharp corners, edges, and conical points is a historically refractory problem. On such regions, the solutions to both the differential equations and their corresponding boundary integral equations turn out to be singular. In this talk, I observe that, when Laplace's equation in three dimensions is formulated as a second kind integral equation of potential theory, the solution (to the integral equation) in the vicinity of a corner is representable by a series of elementary functions. These functions are derived by using the fact that Laplace's equation is scale invariant to separate variables and reduce the three-dimensional problem on the corner to an associated two-dimensional problem in the cross section of the corner. The resulting expressions lend themselves to the construction of accurate discretization schemes, and could be used in the design of high precision numerical algorithms.