The solution of elliptic partial differential equations on regions with
non-smooth boundaries (edges, corners, etc.) is a notoriously
refractory problem. In this talk, I observe that when the problems are
formulated as boundary integral equations of classical potential
theory, the solutions (of the integral equations) in the vicinity of
corners are representable by series of elementary functions. In
addition to being analytically perspicuous, the resulting expressions
lend themselves to the construction of accurate and efficient numerical
algorithms. The results are illustrated by a number of numerical
examples.