Essential spectrum of the double layer potential on curves with cusps

Abstract The essential spectrum of the double layer potential related to the Dirichlet problem on a domain whose boundary is a star-like curve with cusps is studied in the Lebesgue space setting. First, the solvability of such Dirichlet problem on the unit disc and the related integral representation are analyzed. Applying a reduction to singular integral operators with complex conjugation on star-like curves with cusps, and the theory of Wiener-Hopf operators and Mellin convolution operators, the essential spectrum of the double layer potential is established.