Realizations of Differential Operators on Conic Manifolds with Boundary

Abstract

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L_p-Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index.Comment: 41 pages, 1 figur

BoundedH∞-Calculus for Differential Operators on Conic Manifolds with Boundary

“…The reverse inclusion follows as in the proof of Proposition 4.2 in [4] with the special parametrix from ii).…”

“…This calculus has a corresponding parameter-dependent version, some of whose elements we describe now. For a short presentation see for example [4].…”

“…A T means; for details we refer to Section 3.2 of [4]. First, A is elliptic on int D in the standard sense, i.e.…”

“…In this section we shall give such conditions for the case A T = A T,min . In fact, these conditions are obtained by combining the concept of parameter-ellipticity in Schulze's cone calculus and the observations from Section 3.2 of [4]. …”

“…For the analysis of closed extensions of A on D both the corresponding right-inverse as well as the parametrix were constructed in Lemma 3.4 and Propositions 3.3, 3.7 of [4] relying on results of [12] for boundary value problems on smooth manifolds. Both constructions extend to X ∧ .…”