Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension
Abstract
This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded C1,1 domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive q-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are q>d−2 when the observation set is contained in a flat boundary patch and q>d−1−cd+1 on a general C1,1 boundary, where cd+1>0 depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive q-dimensional Hausdorff content for some q>d−1, yielding logarithmic stability from full-time observations for general sources in H01(Ω) and Hölder stability from terminal-time observations for sources in a suitable spectral Gevrey class.
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BibTeX
@misc{li2026inverse,
title = {{Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension}},
author = {Ze Li and Zhiyuan Li and Zhi Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10071},
eprint = {2609.10071},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10071},
}
RIS
TY - UNPB
TI - Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension
AU - Ze Li
AU - Zhiyuan Li
AU - Zhi Zhou
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10071
UR - https://arxiv.org/abs/2609.10071
AB - This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded C^1,1 domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive q-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are q>d−2 when the observation set is contained in a flat boundary patch and q>d−1−c_d+1 on a general C^1,1 boundary, where c_d+1>0 depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive q-dimensional Hausdorff content for some q>d−1, yielding logarithmic stability from full-time observations for general sources in H_0^1(Ω) and Hölder stability from terminal-time observations for sources in a suitable spectral Gevrey class.
ER -