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Weighted heat traces of the Dirichlet Laplacian on Lipschitz domains
Abstract
We prove an asymptotic expansion of a weighted heat trace of the Dirichlet Laplacian on a bounded Lipschitz domain. The weight is given by a power of the distance to the boundary times a bounded function that is continuous near the boundary. Depending on the exponent of the weight, we obtain one-term or two-term asymptotics. We present two applications, namely a version of one-term asymptotics of weighted Riesz means for arbitrary orders, as well as convergence results for a family of measures that describes localization of Laplace eigenfunctions.
Related
- Tangential and normal traces for extension domains with non-Lipschitz boundaries
- Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains
- Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains
- Optimal Lipschitz Thresholds for Weighted Dirichlet Problems, with an Explicit Bessel-Zero Bound for Power Weights
- On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
BibTeX
@misc{kersten2026weighted,
title = {{Weighted heat traces of the Dirichlet Laplacian on Lipschitz domains}},
author = {Lucas Kersten},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16746},
eprint = {2609.16746},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16746},
}
RIS
TY - UNPB
TI - Weighted heat traces of the Dirichlet Laplacian on Lipschitz domains
AU - Lucas Kersten
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16746
UR - https://arxiv.org/abs/2609.16746
AB - We prove an asymptotic expansion of a weighted heat trace of the Dirichlet Laplacian on a bounded Lipschitz domain. The weight is given by a power of the distance to the boundary times a bounded function that is continuous near the boundary. Depending on the exponent of the weight, we obtain one-term or two-term asymptotics. We present two applications, namely a version of one-term asymptotics of weighted Riesz means for arbitrary orders, as well as convergence results for a family of measures that describes localization of Laplace eigenfunctions.
ER -