[論文レビュー] Global Hölder Solvability of parabolic equations on domains with capacity density conditions

Takanobu Hara|arXiv (Cornell University)|Jan 6, 2026

Nonlinear Partial Differential Equations被引用数 0

ひとこと要約

The paper proves global Hölder continuity and existence of weak solutions for linear parabolic equations in divergence form on domains satisfying capacity density conditions, even with data having near-inverse-square boundary singularities.

ABSTRACT

We investigate the Cauchy-Dirichlet problem for linear parabolic equations in divergence form. Under mild assumptions on the source term and the domain, we prove the existence of globally Hölder continuous solutions. Notably, our results accommodate data exhibiting singularities nearly as critical as the inverse square of the distance from the boundary.

研究の動機と目的

• Motivate and address global regularity for parabolic equations in rough domains.
• Establish Hölder continuity of solutions under minimal domain and data assumptions.
• Develop barrier functions (supersolutions) tailored to domains with capacity density conditions.
• Extend the theory to inhomogeneous boundary value problems and boundary regularity.

提案手法

• Formulate the Cauchy-Dirichlet problem for div-form parabolic equations with bounded, uniformly elliptic A(x,t).
• Introduce capacity density condition (CDC) on the spatial domain and derive quantitative boundary Hölder estimates (Lemma 3.9).
• Construct barrier functions via an Ancona-inspired gluing of supersolutions to control boundary behavior (Theorem 4.1).
• Use the barrier together with comparison principles to obtain global Hölder regularity (Theorem 5.5).
• Provide existence and regularity results for f in local L1 with near-boundary singularity allowed (Theorem 5.4, 5.5).
• Extend results to inhomogeneous boundary data and compatibility conditions (Theorem 6.1, 6.2).

実験結果

リサーチクエスチョン

• RQ1 Under what mild conditions on the source term f and the domain D can one guarantee a globally Hölder continuous weak solution to the parabolic problem?
• RQ2 How does the capacity density condition influence boundary regularity and what quantitative estimates can be derived near the boundary?
• RQ3 Can data with strong boundary singularities (nearly inverse-square in distance to boundary) be accommodated in global Hölder solvability?
• RQ4 How can barrier functions be constructed to handle rough domains and be used to obtain global estimates?
• RQ5 What are the implications for inhomogeneous boundary value problems and compatibility conditions?

主な発見

• There exists a globally Hölder continuous weak solution to the parabolic Cauchy-Dirichlet problem on domains satisfying the capacity density condition (CDC).
• A barrier function sΓ is constructed that satisfies HsΓ ≥ cH sΓ / δΓ^2 and bounds δΓ^αH up to constants, enabling boundary control.
• The results allow data with singularity nearly as strong as the inverse square of the distance to the boundary.
• An explicit boundary Hölder estimate is established under the CDC, enabling global Hölder regularity up to the parabolic boundary.
• There are unique weak solutions with Hölder estimates when fδ^{2−α} ∈ L∞, and the theory extends to inhomogeneous boundary data with compatibility conditions.
• The work provides an existence theory for Hölder continuous solutions and extends to boundary value problems (Theorems 5.5, 6.1, 6.2).