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Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains

Benny Avelin, Mingyi Hou

TL;DR

The paper addresses the Dirichlet problem for the stationary Kramers-Fokker-Planck operator $\mathscr{L}$ in bounded domains with rough coefficients, focusing on weak and Perron-Wiener-Brelot (PWB) solutions and the trace issue. It combines variational methods and vanishing-viscosity approximations to establish existence and uniqueness of weak solutions, and shows a renormalized Green's formula holds under a weak trace, enabling a robust comparison principle. Building on these foundations, it develops a full Perron theory in arbitrary bounded domains, proving resolutivity and providing barrier-based boundary regularity results, including that balls and velocity-symmetric domains are regular. The results advance the understanding of boundary value problems for hypoelliptic kinetic equations with rough data, offering tools for further potential-theoretic analysis and applications to metastability and exit-time problems.

Abstract

In this paper, we investigate weak solutions and Perron-Wiener-Brelot solutions to the linear stationary Kramers-Fokker-Planck equation in bounded domains. We establish the existence of weak solutions in product domains by applying the Lions-Lax-Milgram theorem and the vanishing viscosity method. Furthermore, we show that these solutions coincide in well-behaved domains. Building on the existence of weak solutions in product domains, we develop the foundational theory of Perron-Wiener-Brelot solutions in arbitrary bounded domains. Our results rely on recent advancements in the theory of kinetic Fokker-Planck equations with rough coefficients.

Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains

TL;DR

The paper addresses the Dirichlet problem for the stationary Kramers-Fokker-Planck operator in bounded domains with rough coefficients, focusing on weak and Perron-Wiener-Brelot (PWB) solutions and the trace issue. It combines variational methods and vanishing-viscosity approximations to establish existence and uniqueness of weak solutions, and shows a renormalized Green's formula holds under a weak trace, enabling a robust comparison principle. Building on these foundations, it develops a full Perron theory in arbitrary bounded domains, proving resolutivity and providing barrier-based boundary regularity results, including that balls and velocity-symmetric domains are regular. The results advance the understanding of boundary value problems for hypoelliptic kinetic equations with rough data, offering tools for further potential-theoretic analysis and applications to metastability and exit-time problems.

Abstract

In this paper, we investigate weak solutions and Perron-Wiener-Brelot solutions to the linear stationary Kramers-Fokker-Planck equation in bounded domains. We establish the existence of weak solutions in product domains by applying the Lions-Lax-Milgram theorem and the vanishing viscosity method. Furthermore, we show that these solutions coincide in well-behaved domains. Building on the existence of weak solutions in product domains, we develop the foundational theory of Perron-Wiener-Brelot solutions in arbitrary bounded domains. Our results rely on recent advancements in the theory of kinetic Fokker-Planck equations with rough coefficients.
Paper Structure (24 sections, 31 theorems, 125 equations, 1 figure)

This paper contains 24 sections, 31 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The trace problem
  3. Novelty and contributions
  4. Main results
  5. Existence and uniqueness of weak solutions
  6. Perron's method
  7. Preliminaries
  8. Harnack's inequality and Hölder continuity
  9. Strong maximum principle
  10. Renormalized Green's formula and the proof of \ref{['thm:comparison']}
  11. Renormalized Green's formula
  12. Proof of \ref{['thm:comparison']}
  13. Boundary regularity
  14. Existence of weak solutions
  15. Lions-Lax-Milgram theorem
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let $\mathcal{D}\subset\mathbb{R}^{2n}$ satisfy assump:D0 where $\partial\mathcal{U}$ is $\mathrm{C}^{1,1}$, then there is a linear operator such that, for $\varphi\in \mathrm{C}^1(\overline{\mathcal{D}})$, $\varphi|_{\partial^v\mathcal{D}} = 0$, we have the following integration by parts formula:

Figures (1)

  • Figure :

Theorems & Definitions (68)

  • Lemma 3.1: Weak trace
  • Remark 3.2
  • Definition 3.3: Weak solution
  • Definition 3.4
  • Theorem 1
  • Definition 3.5
  • Theorem 2
  • Theorem 3: Comparison principle
  • Remark 3.6
  • Definition 3.7
  • ...and 58 more