Analysis and boundary value problems on singular domains: An approach via bounded geometry

Abstract

We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains that are conformally equivalent to manifolds with boundary and bounded geometry. Our assumptions are satisfied by the domains with a smooth set of singular cuspidal points, and hence our results apply to the class of domains with isolated oscillating conical singularities. In particular, our results generalize the classical L 2 -well-posedness result of Kondratiev for the Laplacian on domains with conical points. However, our domains and coefficients are too general to allow for singular function expansions of the solutions similar to the ones in Kondratiev's theory. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier geometric and analytic results on such manifolds.

Semiclassical pseudodifferential operators and resolvent parametrices on manifolds with ends

“…Analysis on manifolds with bounded geometry is studied by Engel [10], Große and Schneider [11], and Shubin [22]. The study by Ammann, Große and Nistor [1] employed the analysis on manifolds with bounded geometry to that on singular domains.…”

Maximal regularity of parabolic transmission problems

“…Thus, for example, (H m , g m ) is not a urR manifold. A Riemannian manifold with boundary is a urR manifold iff it has bounded geometry in the sense of Schick [35] (also see [17][18][19]26] for related definitions). Detailed proofs of these equivalences can be found in [15].…”

Linear parabolic equations with strong boundary degeneration

“…A Riemannian manifold with boundary is a urR manifold iff it has bounded geometry in the sense of Th. Schick [28] (also see [10], [11], [12], [17] for related definitions). Detailed proofs of these equivalences will be found in [9].…”