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Wiener-type Criterion for the Removability of the Fundamental Singularity for the Heat Equation and its Consequences

Ugur G. Abdulla

TL;DR

This work introduces a Wiener-type criterion for the removability of the fundamental singularity of the heat equation by linking it to parabolic fine topology, $h$-parabolic capacity, and the behavior of the singular parabolic Dirichlet problem. Via the Appell transformation, the authors connect the $h$-parabolic framework on $oldsymbol{ m R}^{N+1}_+$ with the $ ilde{h}$-framework on $oldsymbol{ m R}^{N+1}_-$, establishing a duality between capacity-based, topological, and probabilistic characterizations. The main result proves that removability, uniqueness of the singular PDP, non-thinness in the parabolic minimal-fine topology, and the divergence of a specific capacity-series are all equivalent; analogous statements hold for the point at infinity. The paper also develops geometric and analytic tools—$h$-heat balls, averaging properties, and Harnack-type estimates—that underpin the potential-theoretic approach and facilitate broader applications to parabolic and elliptic PDEs.

Abstract

We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context the criterion determines whether the $h$-parabolic measure of the singularity point is null or positive. From the probabilistic point of view, the criterion presents an asymptotic law for conditional Brownian motion. In {\it U.G. Abdulla, J Math Phys, 65, 121503 (2024)} the Kolmogorov-Petrovsky-type test was established. Here we prove a new Wiener-type criterion for the "geometric" characterization of the removability of the fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of the complementary set near the singularity point.

Wiener-type Criterion for the Removability of the Fundamental Singularity for the Heat Equation and its Consequences

TL;DR

This work introduces a Wiener-type criterion for the removability of the fundamental singularity of the heat equation by linking it to parabolic fine topology, -parabolic capacity, and the behavior of the singular parabolic Dirichlet problem. Via the Appell transformation, the authors connect the -parabolic framework on with the -framework on , establishing a duality between capacity-based, topological, and probabilistic characterizations. The main result proves that removability, uniqueness of the singular PDP, non-thinness in the parabolic minimal-fine topology, and the divergence of a specific capacity-series are all equivalent; analogous statements hold for the point at infinity. The paper also develops geometric and analytic tools—-heat balls, averaging properties, and Harnack-type estimates—that underpin the potential-theoretic approach and facilitate broader applications to parabolic and elliptic PDEs.

Abstract

We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context the criterion determines whether the -parabolic measure of the singularity point is null or positive. From the probabilistic point of view, the criterion presents an asymptotic law for conditional Brownian motion. In {\it U.G. Abdulla, J Math Phys, 65, 121503 (2024)} the Kolmogorov-Petrovsky-type test was established. Here we prove a new Wiener-type criterion for the "geometric" characterization of the removability of the fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of the complementary set near the singularity point.
Paper Structure (8 sections, 14 theorems, 116 equations)

This paper contains 8 sections, 14 theorems, 116 equations.

Table of Contents

  1. Prelude
  2. Overture: Wiener-type Criterion for the Removability of the Fundamental Singularity
  3. Formulation of Problems
  4. Thinness in Parabolic Minimal-Fine Topology
  5. The Main Results
  6. Preliminary Results
  7. Proofs of Theorems \ref{['main theorem']} and \ref{['main theorem2']}
  8. Geometric features of $h$-heat balls, averaging property of $h$-parabolic functions, and Harnack estimates

Key Result

Theorem 1.1

$u_*\equiv 0$ or $u_*>0$, that is to say the fundamental singularity is removable or non-removable according as the series diverges or converges. An equivalent characterization is valid if the series wtest is replaced with

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 6 more