Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems

Abstract

The theory of selfadjoint extensions of symmetric operators, and more generally the theory of extensions of dual pairs, was implemented some years ago for boundary value problems for elliptic operators on smooth bounded domains. Recently, the questions have been taken up again for nonsmooth domains. In the present work we show that pseudodifferential methods can be used to obtain a full characterization, including Kreĭn resolvent formulas, of the realizations of nonselfadjoint second-order operators on $C^{\frac{3}{2}+\epsilon }$ domains; more precisely, we treat domains with $B_{p,2}^{\frac{3}{2}}$-smoothness and operators with $H_q^1$-coefficients, for suitable $p>2(n-1)$ and $q>n$. The advantage of the pseudodifferential boundary operator calculus is that the operators are represented by a principal part and a lower-order remainder, leading to regularity results; in particular we analyze resolvents, Poisson solution operators and Dirichlet-to-Neumann operators in this way, also in Sobolev spaces of negative order.