Dirichlet Heat Kernel for the Laplacian in a Ball

Abstract

We provide sharp two-sided estimates on the Dirichlet heat kernel k1(t, x, y) for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel k1(t, x, y) in terms of its global two-sided sharp estimates.

Sharp estimates of the spherical heat kernel

“…for 1 ≤ j < k. Combining this with Lemma 3.2 (a), we see that the kth term of the sum in (8) dominates if v is large enough. This implies the lemma.…”

“…The example of S d shows that this is a difficult problem even for basic and regular Riemannian manifolds. In this connection, it is perhaps worth mentioning the recent papers [3,4,8,9] where such results were obtained for Dirichlet heat kernels related to Bessel operators in half-lines, the Dirichlet heat kernel in Euclidean balls of arbitrary dimension, and the Fourier-Bessel heat kernel on the interval (0, 1). This was achieved by a clever combination of probabilistic and analytic methods.…”

Asymptotic behaviour of the Bessel heat kernels

“…These results are only qualitatively sharp, which means that exponential terms in upper and lower bound are different. Notice that estimates of the Dirichlet heat kernels for Laplacian in a ball, with the same exponents in upper and lower bounds (so-called sharp estimates), were obtained very recently by Małecki and Serafin in [13]. For more information (including more general second-order differential operators and domains e.g.…”

“…and then we estimate separately each of these integrals, starting with the part J (13) and (14)] q (μ) x (s)…”

Feeling boundary by Brownian motion in a ball

“…Research on short-time boundary behaviour of Dirichlet heat kernels has a long history, but concerns mainly estimates, and not asymptotics (see, among others, [3,5,7,8,12,21,22,17]).…”