Heat Kernel Estimates On Inner Uniform Domains
We introduce conditions on the symmetric and skew-symmetric parts of timedependent, local, regular forms that imply a parabolic Harnack inequality for appropriate weak solutions of the associated heat equation, under natural assumptions on the underlying space. In particular, these local weak solutions are locally bounded and H¨lder continuous. Precise two-sided heat kernel estimates are deo rived from this parabolic Harnack inequality. For Dirichlet forms satisfying our conditions we prove a scale-invariant boundary Harnack principle in inner uniform domains. Inner uniformity is a condition on the boundary of the domain that is described solely in terms of the intrinsic length metric of the domain. In addition, we show that the Martin boundary of an inner uniform domain is homeomorphic to the boundary of the domain with respect to its completion in the inner distance. The main result of this work are two-sided Gaussian bounds for Dirichlet heat kernels corresponding to (non-)symmetric, local, regular Dirichlet forms. These bounds hold in domains that satisfy the inner uniformity condition. The proof uses the parabolic Harnack inequality and the boundary Harnack principle described above, as well as the Doob h-transform technique. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa, A. Ancona, P. Gyrya, L. Saloff-Coste and K.-T. Sturm.
parabolic Harnack inequality; boundary Harnack principle; heat kernel
Saloff-Coste, Laurent Pascal
Strichartz, Robert Stephen; Cao, Xiaodong
Ph.D. of Mathematics
Doctor of Philosophy
dissertation or thesis