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The regularity problem with a weaker condition on only the transversal direction

Martin Ulmer

TL;DR

The paper studies the regularity boundary value problem for the elliptic operator $L=\mathrm{div}(A\nabla\cdot)$ on the upper half-space when the transversal variation is mild. It introduces a weaker mixed $L^1-L^\infty$ condition on $\partial_t A$ together with a growth bound $|\partial_t A|t\le C$ and proves that the elliptic measure $\omega$ lies in $A_\infty(d\sigma)$, yielding $L^p$ solvability for some $p>1$ of the Dirichlet problem; this also provides an alternative route to the $t$-independent case via integration by parts and Kato-conjecture tools. The authors develop a comprehensive framework based on the semigroup $\mathcal P_t$, kernel/off-diagonal estimates, Carleson and tent-space techniques, and Hardy-atom analysis to lift $L^2$-type control to $L^p$-control through real interpolation. The result broadens the class of coefficients for which regularity is solvable in the unbounded upper half-space and connects to the well-known $A_\infty$-solvability paradigm for elliptic boundary value problems, offering a robust, non-potential-based approach. Overall, the work advances the understanding of how transversal coefficient variation impacts boundary regularity and Dirichlet solvability in elliptic theory.

Abstract

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ condition that only depends on $\partial_t A$, the derivative of $A$ in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem.

The regularity problem with a weaker condition on only the transversal direction

TL;DR

The paper studies the regularity boundary value problem for the elliptic operator on the upper half-space when the transversal variation is mild. It introduces a weaker mixed condition on together with a growth bound and proves that the elliptic measure lies in , yielding solvability for some of the Dirichlet problem; this also provides an alternative route to the -independent case via integration by parts and Kato-conjecture tools. The authors develop a comprehensive framework based on the semigroup , kernel/off-diagonal estimates, Carleson and tent-space techniques, and Hardy-atom analysis to lift -type control to -control through real interpolation. The result broadens the class of coefficients for which regularity is solvable in the unbounded upper half-space and connects to the well-known -solvability paradigm for elliptic boundary value problems, offering a robust, non-potential-based approach. Overall, the work advances the understanding of how transversal coefficient variation impacts boundary regularity and Dirichlet solvability in elliptic theory.

Abstract

We study an elliptic operator on the upper half space. It is known that if the matrix is independent in the transversal -direction, then the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the -independence condition by introducing a mixed condition that only depends on , the derivative of in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem.

Paper Structure

This paper contains 10 sections, 20 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and Setting
  4. The approximation operator $\mathcal{P}_t$
  5. Kernel bounds and $L^2-L^2$ off-diagonal estimates
  6. $L^2$ Square Function Bounds on terms involving $\mathcal{P}_t$
  7. Bounds for Carleson functions
  8. Square function bounds on Hardy atoms
  9. Tent spaces and real interpolation
  10. Proof of Theorem \ref{['MAINTHM']}

Key Result

Theorem 1.4

Assume $L:=\mathrm{div}(A\nabla \cdot)$ is uniformly elliptic operator with bounded, merely measurable coefficients and let $\Omega=\mathbb{R}^{n+1}_+$. If $A$ satisfies $|\partial_t A|\leq C/t$ and cond:mixedL1LInftyCond then there exists $p>1$ such that eq:RPinequalityElliptic holds and hence the

Theorems & Definitions (28)

  • Theorem 1.4
  • Definition 2.3: $(RP)_p^L$
  • Proposition 2.5: Prop 4.3 in hofmann_lp_2022 or Theorem 6.17 in ouhabaz_analysis_2004
  • Proposition 2.7: Prop 11 in hofmann_dirichlet_2022 and Cor. 5.6 in ulmer_mixed_2024, proof of Lemma 6.4 in ulmer_mixed_2024
  • Corollary 2.11
  • Proposition 2.12: Off-diagonal estimates
  • Proposition 2.13: Lemma 4.1 and Lemma 4.2 in ulmer_mixed_2024
  • Proposition 2.16: Lemma 7.7 in ulmer_mixed_2024
  • Proposition 2.19: Lemma 7.2 and Lemma 7.3 in ulmer_mixed_2024
  • Lemma 2.22
  • ...and 18 more