The Spectral Radius of the Classical Layer Potentials on Convex Domains

Abstract

Let D denote a bounded Lipschitz domain in R ⁿ. For almost every (with respect to surface measure dσ)Q ∈∂D the exterior normal N _Q at Q exists. The solution u to the Dirichlet problem,

$$\Delta u = 0 \text{ i}n \ \ D, \ \ u \vert_{\partial D} = g$$

, with g ∈ L ²(∂D,dσ) can be represented in the form of the classical double layer potential

$$u(X) = \frac{1}{\omega_n} \int\limits_{\partial D} \frac{N_Q\circ(Q - X)}{\left\vert X - Q \right\vert^n}[((1/2)I + K)^{-1}g](Q)d\sigma(Q)$$

.

Partially supported by NSF Grant Number DMS 90-01411.