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Inverse problems and continuity of heat content coefficients

  • Ryoki Endo
  • Katie Gittins
  • Jacob Walker
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Abstract

We consider the small-time asymptotic expansions for the Dirichlet and Open heat contents of convex planar domains. We prove that the small-time asymptotic expansion for the Open heat content can determine a triangle among triangles, a parallelogram among parallelograms, and an acute trapezium among acute trapezia of the same height. In addition, for a sequence of convex polygons converging to a convex set with smooth boundary in the sense of Hausdorff, we show that the coefficients of order t in the small-time asymptotic expansions of the Dirichlet and the Open heat contents converge to those of the limiting set. While, for a sequence of convex sets with smooth boundary converging to a convex polygon in the sense of Hausdorff, we show that the coefficients of order t do not converge to that of the limiting polygon.

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Cite

BibTeXRIS
BibTeX
@misc{endo2026inverse,
  title = {{Inverse problems and continuity of heat content coefficients}},
  author = {Ryoki Endo and Katie Gittins and Jacob Walker},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32613},
  eprint = {2610.32613},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32613},
}
RIS
TY  - UNPB
TI  - Inverse problems and continuity of heat content coefficients
AU  - Ryoki Endo
AU  - Katie Gittins
AU  - Jacob Walker
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32613
UR  - https://arxiv.org/abs/2610.32613
AB  - We consider the small-time asymptotic expansions for the Dirichlet and Open heat contents of convex planar domains. We prove that the small-time asymptotic expansion for the Open heat content can determine a triangle among triangles, a parallelogram among parallelograms, and an acute trapezium among acute trapezia of the same height. In addition, for a sequence of convex polygons converging to a convex set with smooth boundary in the sense of Hausdorff, we show that the coefficients of order t in the small-time asymptotic expansions of the Dirichlet and the Open heat contents converge to those of the limiting set. While, for a sequence of convex sets with smooth boundary converging to a convex polygon in the sense of Hausdorff, we show that the coefficients of order t do not converge to that of the limiting polygon.
ER  -