A heat equation approach to some boundary values problems in analysis and geometry

A basic problem in analysis is the Dirichlet problem for the Laplacian in the upper half-space. It is a classical fact that the trace of the Neumann derivative of the solution equals the half-Laplacian of the boundary datum. In a seminal 2007 paper Caffarelli and Silvestre considerably revived the interest in a version of this problem which has several ramifications. A remarkable feature of their so-called extension problem  is that, similarly to its classical predecessor, it is possible to recover the fractional Laplacian of the boundary datum as a weighted Neumann trace of the solution. This allows to convert problems in which the leading operator is the nonlocal fractional Laplacian into ones involving the (local) extension partial differential operator. One important application of this feature is, for instance, in the theory of free boundaries, when one looks at lower-dimensional obstacle problems of Signorini type. But the extension problem is also of interest in geometry, in the study of scattering eigenvalue problems in real hyperbolic space.  

In this lecture I present an approach, based on the heat equation and some of its variants, to some extension problems arising in conformal geometry.  One of the highlights will be the construction of the heat kernel of an extension problem introduced by Frank, Del Mar Gonzalez, Monticelli and Tan. Such problem plays a pervasive role in conformal CR geometry and, among other things, it serves as a way to define the fractional powers of the conformal sublaplacian on the Heisenberg group. I will discuss some intertwining formulas related to conformal geometry that are instrumental in inverting the relevant nonlocal operators, as well as in constructing explicit solutions of some nonlocal Yamabe problems. 

The leitmotif of my talk will be emphasising the so far unexplored connection of the relevant hybrid equations with the heat kernel of the generalised operator of Ornstein-Uhlenbeck type in the opening of H\"ormander's groundbreaking 1967 work on hypoellipticity. Effort will be made to provide a self-contained non-technical overview.