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arXiv · 2607.18784

Short-time behavior and invariant surfaces for caloric functions with non-constant boundary values

Diego Berti·Rolando Magnanini·Michele Marini
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Abstract

We consider a Cauchy-Dirichlet problem for a heat equation with variable coefficients in non-divergence form. The initial values are assumed to be homogeneous, while the Dirichlet boundary values are non-constant. In this setting, we derive an asymptotic formula for the short-time behavior of the solution, which extends the celebrated Varadhan formula. We stress the fact that the boundary values are allowed to vanish or change sign. The formula is obtained by combining the original ideas of Varadhan with those of Evans and Ishii, pertaining to the theory of viscosity solutions, and some further remarks. In passing, we prove the elliptic counterpart of the formula. This concerns the slow-diffusion behavior of the solutions of the resolvent equation. Furthermore, for the case of the classical Laplace operator, we prove asymptotic formulas for the heat content and the mean value of the solution of the resolvent equation on spheres touching the boundary of the domain. These extend to the case of non-constant boundary values, certain formulas previously obtained by the second author and S. Sakaguchi. The formulas involve the principal curvatures at the touching point and the Dirichlet boundary values. We then use our formulas to describe time-invariant surfaces for the heat equation, in presence of non-constant Dirichlet boundary values. We give two symmetry results in case of non-negative Dirichlet boundary values and a non-existence result, when those values change sign.

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Diego Berti, Rolando Magnanini, Michele Marini. 2026-07-21. Short-time behavior and invariant surfaces for caloric functions with non-constant boundary values. https://arxiv.org/abs/2607.18784

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