Extensions theory and Kreı̆n-type resolvent formulas for nonsmooth boundary value problems

For a strongly elliptic second-order operator A on a bounded domain Ω ⊂ R it has been known for many years how to interpret the general closed L2(Ω)-realizations of A as representing boundary conditions (generally nonlocal), when the domain and coe cients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coe cients, including the case of Hölder C 3 2-smoothness, in such a way that pseudodi erential methods are still available for resolvent constructions and ellipticity considerations. We show how it can be done for domains with B 3 2 2,p-smoothness and operators with H 1 q -coe cients, for suitable p > 2(n − 1) and q > n. In particular, Kre n-type resolvent formulas are established in such nonsmooth cases. Some unbounded domains are allowed.