The Resolvent of Closed Extensions of Cone Differential Operators

Abstract

Abstract. We study closed extensions A of an elliptic differential operator A on a manifold with conical singularities, acting as an unbounded operator on a weighted L p -space. Under suitable conditions we show that the resolvent (λ − A) −1 exists in a sector of the complex plane and decays like 1/|λ| as |λ| → ∞. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A.As an application we treat the Laplace-Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problemu − ∆u = f , u(0) = 0.

Realizations of Differential Operators on Conic Manifolds with Boundary

“…While our final results -given in some detail in (a)-(d) at the end of this sectionlook similar to those in [4,13,23], the analysis in our setting becomes much more difficult. This starts with simple facts: The domains in general are not invariant under multiplication by cut-off functions, since the boundary condition then need not remain fulfilled.…”

“…He worked with weighted L 2 -spaces and showed that, under a natural ellipticity assumption, both the minimal and the maximal extension are Fredholm operators and that the quotient D max /D min of their domains is finitedimensional; its dimension can be computed from symbol data of A (more precisely, from the meromorphic structure of the conormal symbol). Gil and Mendoza [4], Proposition 3.6, obtained an improved description of D min , while the structure of the maximal domain was studied in Schrohe and Seiler [23], Theorem 2.8 (also in case p = 2). For similar considerations see also Liu and Witt [14].…”

Resolvents of cone pseudodifferential operators, asymptotic expansions and applications

“…In the setting of resolvents of close extensions of a cone differential operator, there are recent results by Schrohe and Seiler [35], Falomir et al [11], and Falomir et al [10]. More recently, Gil et al [18] proved the existence of the resolvent and sectors of minimal growth for the closed extensions of a general cone differential operator.…”

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

“…[15] for a presentation) allows the construction of a parametrix which yields the resolvent to A T,min . Based on the results on resolvents of closed extensions of cone differential operators in [10], [11] by Gil, Krainer, and Mendoza, in [16] by Krainer, and in [22], we expect to obtain such natural conditions also for extensions different from the minimal one. This analysis, however, is not the focus of the present paper.…”