# Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems

Abels, Helmut and Grubb, Gerd and Wood, Ian (2014) Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems. Journal of Functional Analysis, 266 (7). pp. 4037-4100. ISSN 0022-1236. (The full text of this publication is not available from this repository)

For a strongly elliptic second-order operator $A$ on a bounded domain $\Omega\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(\Omega)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of H\"older $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show how it can be done for domains with $B^\frac32_{2,p}$-smoothness and operators with $H^1_q$-coefficients, for suitable $p>2(n-1)$ and $q>n$. In particular, Kre\u\i{}n-type resolvent formulas are established in such nonsmooth cases. Some unbounded domains are allowed.