  We present new integral representations in two dimensions for the elastance
problem in electrostatics and the mobility problem in Stokes flow. These
representations lead to resonance-free Fredholm integral equations of the
second kind and well conditioned linear systems upon discretization. By
coupling our integral equations with high order quadrature and fast multipole
acceleration, large-scale problems can be solved with only modest computing
resources. We also discuss some applications of these boundary value problems
in applied physics.
