No Arabic abstract
This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the results apply to the Dirichlet heat kernel associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable coefficients in inner uniform domains in $mathbb R^n$. The results are applicable to any convex domain, to the complement of any convex domain, and to more exotic examples such as the interior and exterior of the snowflake.
This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.
Heat-kernel expansion and zeta function regularisation are discussed for Laplace type operators with discrete spectrum in non compact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several examples. It is pointed out that for a class of exponential (analytic) interactions, generically the non-compactness of the domain gives rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic continuation of the associated zeta function is investigated. A simple model is considered, for which the analytic continuation of the zeta function is not regular at the origin, displaying a pole of higher order. For a physically meaningful evaluation of the related functional determinant, a generalised zeta function regularisation procedure is proposed.
In the context of a metric measure Dirichlet space satisfying volume doubling and Poincare inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.
For large classes of non-convex subsets $Y$ in ${mathbb R}^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$ exists - despite the fact that the entropy is not semiconvex - and coincides with the heat flow on $Y$ with Neumann boundary conditions.
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushimas ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yaus and Karps Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.