The heat trace for domains with curved corners

We derive the short-time expansion of the heat trace on planar curvilinear polygons through order $t^{1/2}$, with both Dirichlet and Neumann boundary conditions. For such domains we show that the coefficient of $t^{1/2}$ in the expansion splits into a boundary integral of $κ^2$ and a sum of local corner contributions, one for each vertex. Each curved corner contribution depends only on the interior angle $α$ and on the limiting curvatures $κ_{\pm}$ on the adjacent sides. For Dirichlet boundary conditions, using a conformal model and a parametrix construction on the sector heat space, we express this contribution in the form [ \mathcal C_{1/2}(α,κ_+,κ_-) = c_{1/2}(α),\frac{κ_+ + κ_-}{4\sin(α/2)}, ] where $c_{1/2}(α)$ is given by the Hadamard finite part of an explicit trace over the exact sector. For right angles we compute $c_{1/2}(π/2)=\sqrt{2}/(16\sqrtπ)$, which yields the closed formula [ \mathcal C_{1/2}(π/2,κ_+,κ_-)=\frac{κ_+ + κ_-}{32\sqrtπ}. ] Combined with previous work on the coefficients of $t^{-1}$, $t^{-1/2}$ and $t^0$ for such domains, this determines the remaining local heat trace coefficient for planar curvilinear polygons up to order $t^{1/2}$. As an inverse spectral application we extend a result in the literature by showing that any admissible curvilinear polygon that is Dirichlet isospectral to a polygon must itself be a polygon with straight sides.

💡 Research Summary

The paper addresses a long‑standing gap in the short‑time heat‑trace asymptotics for planar domains whose boundaries are piecewise smooth but may contain curved corners (so‑called curvilinear polygons). While the coefficients of the $t^{-1}$, $t^{-1/2}$ and $t^{0}$ terms have been known for smooth domains and later extended to curvilinear polygons, the next term, proportional to $t^{1/2}$, had remained unknown because it is the first term where the interaction between boundary curvature and corner geometry appears.

The authors first prove that the $t^{1/2}$ coefficient splits naturally into two parts: an integral over the smooth portions of the boundary of the square of the curvature, $\int_{\partial\Omega}\kappa^{2},ds$, and a sum over the vertices of a purely local “curved‑corner contribution’’ $ \mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$. Here $\alpha$ is the interior angle at the vertex and $\kappa_{\pm}$ are the limiting curvatures of the two adjacent sides.

For Dirichlet boundary conditions they construct a conformal model of a single curved corner and compute the contribution by a parametrix construction on the sector heat space. The result is the factorisation
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