Ventcel’ boundary value problems for elliptic Waldenfels operators

Abstract

In this paper we study a class of first-order Ventcel’ boundary value problems for second-order, elliptic Waldenfels integro-differential operators. More precisely, by using real analysis techniques such as Strichartz norms and the complex interpolation method we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces of \(L^{p}\) type which extend earlier theorems due to Bony–Courrège–Priouret and Runst–Youssfi to the general degenerate case. Our proof is based on various maximum principles for second-order, elliptic Waldenfels operators with discontinuous coefficients in the framework of \(L^{p}\) Sobolev spaces.