具有弯曲角的区域的热迹
The heat trace for domains with curved corners
AI总结：
研究了具有弯曲角的平面多边形的热迹，发现其在阶$ t^{1/2} $处首次出现角和曲率的相互作用，并证明了Dirichlet角因子的符号法则，表明其符号仅由角是否为凸或凹决定。
摘要：
平面多边形的热迹包含仅依赖于角开口角的角项，而光滑平面区域的热迹包含边界上的曲率项。我们证明，对于曲边多边形，这两种现象首次在阶$ t^{1/2} $处相互作用。我们计算了这一首个角-曲率热不变量，并证明了其Dirichlet角因子的符号法则：其符号仅由角是否为凸或凹决定。更具体地说，我们推导了在Dirichlet和Neumann边界条件下的局部热迹展开式，直到阶$ t^{1/2} $。新的系数分解为通常光滑边界贡献和一个依赖于内部角$ α $和相邻弧的一侧极限曲率$ κ_{±} $的局部曲边角项之和。在Dirichlet情况下，曲边角贡献的形式为$ C_{1/2}(α,κ_+,κ_-) = c_{1/2}(α) rac{κ_+ + κ_-}{4\sin(α/2)} $，其中$ c_{1/2}(α) $由一个显式的扇形热核积分给出。我们确定了其对于每个$ 0<α<2π $的符号。符号法则具有谱后果：它给出了曲边多边形与直边多边形Dirichlet等谱的新的障碍。特别是，每一个Dirichlet等谱于直边多边形的曲边多边形本身必须是直边的，从而去除了Enciso和Gómez-Serrano定理中对直角的假设。
英文摘要：
The heat trace of a planar polygon contains corner terms depending only on the opening angles, while the heat trace of a smooth planar domain contains curvature terms along the boundary. We show that, for curvilinear polygons, these two phenomena first interact at order $t^{1/2}$. We compute this first corner-curvature heat invariant and prove a sharp sign law for its Dirichlet angular factor: its sign is determined solely by whether the corner is convex or reflex. More precisely, we derive the local heat trace expansion through order $t^{1/2}$, for both Dirichlet and Neumann boundary conditions. The new coefficient decomposes into the usual smooth-boundary contribution and a sum of local curved-corner terms, each depending only on the interior angle $α$ and the one-sided limiting curvatures $κ_{\pm}$ of the adjacent arcs. In the Dirichlet case, the curved-corner contribution has the form $C_{1/2}(α,κ_+,κ_-) = c_{1/2}(α)\frac{κ_+ + κ_-}{4\sin(α/2)}$, with $c_{1/2}(α)$ given by an explicit sector heat kernel integral. We determine its sign for every $0<α<2π$. The sign law has a spectral consequence: it gives a new obstruction to a curvilinear polygon being Dirichlet-isospectral to a straight-sided polygon. In particular, every convex curvilinear polygon Dirichlet-isospectral to a straight-sided polygon must itself be straight-sided, removing the assumption of straight corners from the theorem of Enciso and Gómez-Serrano.
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