  We investigate the behavior of integral formulations of variable coefficient
elliptic partial differential equations (PDEs) in the presence of steep
internal layers. In one dimension, the equations that arise can be solved
analytically and the condition numbers estimated in various L_p norms. We show
that high-order accurate Nystr\"{o}m discretization leads to well-conditioned
finite dimensional linear systems if and only if the discretization is both
norm-preserving in a correctly chosen L_p space and adaptively refined in the
internal layer.
