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

Technical Summary: Short-Time Behavior and Invariant Surfaces for Caloric Functions with Non-Constant Boundary Values

Problem Statement
This paper investigates the short-time asymptotic behavior of solutions to the Cauchy-Dirichlet problem for heat equations with variable coefficients in non-divergence form. The study focuses on a specific setting where the initial data is homogeneous ($u_0 \equiv 0$), while the Dirichlet boundary values ($f$) are non-constant. The authors address two primary objectives:

1. Deriving asymptotic formulas for the solution $u(x,t)$ and its associated resolvent $U_\varepsilon(x)$ as $t \to 0^+$ and $\varepsilon \to 0^+$, respectively, extending the classical Varadhan formulas to cases where $f$ may vanish or change sign.
2. Utilizing these asymptotic results to analyze the geometry of "time-invariant surfaces" (level surfaces $\Gamma$ where $u(x,t) = c(t)$) in the presence of such non-constant boundary data.

Methodology
The authors employ a combination of techniques from the theory of viscosity solutions, stochastic control, and spectral analysis:

• Viscosity Solutions and Barriers: The derivation of the asymptotic formulas combines Varadhan's original barrier construction methods with the viscosity solution framework developed by Evans and Ishii. This involves a Hopf-Cole transformation ($V_\varepsilon = -\varepsilon \log U_\varepsilon$) to convert the linear elliptic problem into a nonlinear one related to the eikonal equation.
• Compactness and Regularization: The authors establish Arzelà-Ascoli compactness for the family of transformed solutions $\{V_\varepsilon\}$ and utilize convolution techniques to regularize distance functions, proving they act as subsolutions to the relevant equations.
• Laplace Transform Relations: The connection between the parabolic solution $u$ and the elliptic resolvent $U_\varepsilon$ is formalized via a modified Laplace transformation. The authors leverage Tauberian-type arguments (specifically Lemma A.1) to transfer asymptotic results from the elliptic to the parabolic regime.
• Geometric Analysis: For the rigidity results, the paper employs the method of moving planes (Serrin's method), Alexandrov's reflection principle, and the study of principal curvatures. The authors analyze the large-time behavior of solutions to show that the existence of invariant surfaces imposes strong constraints on the boundary data.

Key Contributions and Results

1. Extended Varadhan Formulas
The paper generalizes the classical Varadhan formulas, which relate the short-time behavior of heat kernels to Riemannian distance functions.

• Non-negative Boundary Values: When $f \ge 0$ and not identically zero, the authors prove that the asymptotic behavior is governed by the distance to the set $S = \{x \in \partial\Omega : f(x) > 0\}$. Specifically:
  $$\lim_{t \to 0^+} [-4t \log u(x,t)] = d_S(x)^2$$
  $$\lim_{\varepsilon \to 0^+} [-\varepsilon \log U_\varepsilon(x)] = d_S(x)$$
  where $d_S$ is the intrinsic distance derived from the coefficient matrix $A(x)$.
• Sign-Changing Boundary Values: When $f$ changes sign, the asymptotics depend on the relative distances to the positive set $S_+$ and negative set $S_-$. The limit is determined by the closer set ($S_+$ or $S_-$), provided the distances are distinct. The case where distances are equal ($d_{S_+} = d_{S_-}$) is shown to be delicate, potentially leading to infinite limits or dependence on the specific values of $f$.

2. Asymptotics for Heat Content on Spheres
For the classical Laplace operator ($A(x)=I$), the authors derive new asymptotic formulas for the heat content (and mean value of the resolvent) on spheres touching the boundary. These formulas extend previous results by Sakaguchi and the second author to non-constant boundary values. The limits involve the principal curvatures of the boundary at the touching point and the value of the boundary function $f$ at that point.

3. Rigidity and Symmetry Results
The paper applies the derived formulas to characterize time-invariant surfaces:

• Symmetry for Non-negative Data: If the boundary data is non-negative and a time-invariant surface exists for all time, the domain $\Omega$ and the surface must be concentric balls, and the boundary data $f$ must be constant. This result is achieved by analyzing the large-time behavior, which forces the boundary data to be constant via analyticity.
• Two Invariant Surfaces: If two disjoint time-invariant surfaces exist for short times (without relying on large-time behavior), the domain and surfaces are concentric balls, and $f$ is constant.
• Non-existence for Sign-Changing Data: If the boundary data changes sign and the domain satisfies certain regularity conditions, no time-invariant surface can exist that is compactly contained within the domain (i.e., not touching the boundary). The authors note that invariant surfaces extending to the boundary or with self-intersections may exist in this regime.

Significance and Claims
The paper claims to extend the celebrated Varadhan formulas to a broader class of boundary conditions, specifically addressing the case where boundary values vanish or change sign, a scenario not covered by previous standard results. The authors emphasize that their approach unifies the elliptic (slow-diffusion) and parabolic (short-time) behaviors.

Regarding rigidity, the paper highlights that the presence of non-constant boundary values significantly alters the landscape of time-invariant surfaces. While constant boundary values allow for spherical symmetry, the introduction of sign-changing data generally precludes the existence of compactly contained invariant surfaces. The authors position their work as a necessary step in understanding the "overdetermination" introduced by invariant surfaces in non-homogeneous settings, providing a foundation for future investigations into the delicate geometry of surfaces in sign-changing regimes. The results are presented as rigorous extensions of existing literature on the Matzoh Ball Soup Problem and related rigidity questions.