Laplace Dirichlet heat kernels in convex domains

Grzegorz Serafin

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arXiv · 2110.06147

Laplace Dirichlet heat kernels in convex domains

Grzegorz Serafin

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Abstract

We provide general lower and upper bounds for Laplace Dirichlet heat kernel of convex $\mathcal C^{1,1}$ domains. The obtained estimates precisely describe the exponential behaviour of the kernels, which has been known only in a few special cases so far. Furthermore, we characterize a class of sets for which the estimates are sharp, i.e. the upper and lower bounds coincide up to a multiplicative constant. In particular, this includes sets of the form $\{x\in \R^n: x_n>a|(x_1,...,x_{n-1})|^p\}$ where $p\geq2$, $n\geq2 $ and $a>0$.

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Grzegorz Serafin. 2021-10-12. Laplace Dirichlet heat kernels in convex domains. https://arxiv.org/abs/2110.06147

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