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Boundedness of weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains

Mingyi Hou

TL;DR

The paper addresses the challenge of proving global boundedness up to the boundary for weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains. It develops a De Giorgi iteration adapted to a homogeneous Lie-group framework, combining a renormalization formula with novel $L^1$--$L^{p_0}$ and $L^2$--$L^{2p_0}$ embeddings derived from the fundamental solution. Central to the method are the integrability properties of the Kolmogorov kernel, sharp energy and Caccioppoli estimates for truncations, and a carefully constructed truncation/renormalization scheme that handles boundary traces in the weak sense. The results yield global $L^{\infty}$ bounds and a weak maximum principle under two coefficient regimes, recovering the uniform-parabolic exponents as a special case and establishing optimal exponents for the degenerate hypoelliptic setting. The framework extends the classical boundedness theory to hypoelliptic degenerate equations and opens avenues for nonlinear extensions and broader operator classes within same geometric structure.

Abstract

We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We employ the renormalization formula to handle boundary values and provide energy estimates. An $L^1$-$L^p$ type embedding estimate derived from the fundamental solution is utilized to incorporate lower-order divergence terms. This work naturally extends the boundedness theory for uniformly parabolic equations, with matching exponents for the coefficients.

Boundedness of weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains

TL;DR

The paper addresses the challenge of proving global boundedness up to the boundary for weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains. It develops a De Giorgi iteration adapted to a homogeneous Lie-group framework, combining a renormalization formula with novel -- and -- embeddings derived from the fundamental solution. Central to the method are the integrability properties of the Kolmogorov kernel, sharp energy and Caccioppoli estimates for truncations, and a carefully constructed truncation/renormalization scheme that handles boundary traces in the weak sense. The results yield global bounds and a weak maximum principle under two coefficient regimes, recovering the uniform-parabolic exponents as a special case and establishing optimal exponents for the degenerate hypoelliptic setting. The framework extends the classical boundedness theory to hypoelliptic degenerate equations and opens avenues for nonlinear extensions and broader operator classes within same geometric structure.

Abstract

We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We employ the renormalization formula to handle boundary values and provide energy estimates. An - type embedding estimate derived from the fundamental solution is utilized to incorporate lower-order divergence terms. This work naturally extends the boundedness theory for uniformly parabolic equations, with matching exponents for the coefficients.
Paper Structure (16 sections, 15 theorems, 127 equations)

This paper contains 16 sections, 15 theorems, 127 equations.

Table of Contents

  1. Introduction and the main result
  2. A brief survey of the question
  3. Sketch of the proof and outline
  4. Intergrability, embedding and renormalization
  5. Integrability of the Kolmogorov kernel
  6. Some embedding results
  7. Renormalization of the weak solution
  8. Proof of Theorem 1: the De Giorgi iteration
  9. $\mathrm{L}^{2p_0}$ embedding for truncations of the undercut
  10. $\mathrm{L}^{2p_0}$ energy estimate for the undercut in sufficiently small time cylinder
  11. Caccioppoli estimate for the undercut in sufficiently small time cylinder
  12. Iteration inequalities
  13. Proof of \ref{['eq:linfl2']}
  14. Proof of \ref{['eq:linfdata']}
  15. Iteration under Data 2
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Omega = \mathcal{V} \times \mathcal{U}\subset\mathbb{R}^{m_0}\times\mathbb{R}^{N-m_0}$ be a bounded product domain with $\partial\mathcal{V}$ being $\mathrm{C}^{0,1}$ and $\partial\mathcal{U}$ being $\mathrm{C}^{1,1}$, and let the assumptions assump:ellipticassump:B hold. If $u\in\mathrm{V}^0_ and

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2
  • Lemma 2.1: Integrablity of the fundamental solution
  • proof
  • Lemma 2.2: Young's convolution inequality
  • proof
  • Proposition 2.3: $\mathrm{L}^1$--$\mathrm{L}^p$ embedding
  • ...and 20 more