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

2010/01/01 by Helmut Abels, Abels, Helmut, Gerd Grubb +3
Engineering · Computer Science · #Numerical methods in engineering #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics

paper · pdf · doi:10.48550/arxiv.1008.3281

Abstract

For a strongly elliptic second-order operator A on a bounded domain\n\Ω\⊂ \ℝⁿ it has been known for many years how to interpret\nthe general closed L₂(\Ω)-realizations of A as representing boundary\nconditions (generally nonlocal), when the domain and coefficients are smooth.\nThe purpose of the present paper is to extend this representation to nonsmooth\ndomains and coefficients, including the case of H "older\nC ^frac32+\ε-smoothness, in such a way that pseudodifferential\nmethods are still available for resolvent constructions and ellipticity\nconsiderations. We show how it can be done for domains with\nB^ frac32_p,2-smoothness and operators with H¹_q-coefficients, for\nsuitable p>2(n-1) and q>n. In particular, Kre u in-type resolvent\nformulas are established in such nonsmooth cases. Some unbounded domains are\nallowed.\n

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