Gluing Spaces and Analysis

Abstract

In the first part of this work we study the gluing of metric measure spaces. The gluing is defined by a bilipschitz map which identifies the gluing sets. The new metric is given by the minimal length of all possible paths on the glued space. On each of these spaces a strongly local, regular Dirichlet form is defined. Additionally, each space satisfies a doubling property and a strong scaling invariant Poincaré inequality for all balls holds true. We derive the doubling property and the scaling invariant Poincaré inequality on the glued space provided a lower bound on the "heat transmission coefficient" for certain sets holds true. For the proof only assumptions on the Dirichlet forms on the separate pieces are used.<br /> These results imply upper and lower Gaussian estimates on the heat kernel, short-time asymptotics and the Feller property of the associated process on the glued space.<br /> In the second part we give some generalizations of results by Charles Amick. These are characterizations of the validity of the Poincaré inequality and of Rellichs compact embedding theorem on a domain in terms of a quantity extracted from the boundary. We prove characterizations of this kind for strongly local regular Dirichlet forms on metric measure spaces which satisfy a scaling invariant Poincaré inequality.