Pseudodifferential operators on non-compact manifolds and analysis on polyhedral domains

Abstract

We review the definition of a Lie manifold (M, V) and the construction of the algebra Ψ ∞ V (M ) of pseudodifferential operators on a Lie manifold (M, V). We give some concrete Fredholmness conditions for pseudodifferential operators in Ψ ∞ V (M ) for a large class of Lie manifolds (M, V). These Fredholm conditions have applications to boundary value problems on polyhedral domains and to non-linear PDEs on non-compact manifolds. As an application, we determine the spectrum of the Dirac operator on a manifold with multi-cylindrical ends.

Boundary value problems and layer potentials on manifolds with cylindrical ends

“…See also [39]. Also, see [47] for an extension of the above results to L p -spaces, and [10] for some applications to non-linear evolution equations.…”

Getzler rescaling via adiabatic deformation and a renormalized index formula

“…Under the above conditions on the integrating Lie groupoid, full ellipticity is equivalent to the operator being Fredholm, see e.g. [39]. We will prove Theorem 1.2 in the final section.…”

“…Since e x and r * are surjective it follows that ̺ is surjective. We prove the injectivity using the Hausdorff condition and the assumption that G restricts to the pair groupoid, see also [39]. Let z ∈ M 0 be fixed and denote by e z : S(G) → S(G z ) the evaluation T = (T x ) x∈M → T z .…”

Resolvents of Elliptic Boundary Problems on Conic Manifolds

“…In particular, results about the structure and growth of resolvents of operators with respect to the spectral parameter have immediate consequences as regards the existence, uniqueness, and maximal regularity of solutions to parabolic linear and semilinear equations. Hence this paper naturally belongs to the study of partial differential equations in nonsmooth domains, a subject which due to its importance in models from applications has recently attracted increased attention (see Kapanadze and Schulze, 2003;Kozlov et al, 2001;Maz'ya et al, 2000;Mitrea et al, 2001;Mitrea and Nistor, 2004;Mitrea et al, 2005;Nistor, 2005, to mention only a few). Our results, in particular, give a fairly complete Krainer picture about the existence of sectors of minimal growth for L 2 -based realizations of general elliptic boundary value problems in domains with cone-like singularities on the boundary.…”