Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Meyers exponent rules the first-order approach to second-order elliptic boundary value problems

Pascal Auscher, Tim Böhnlein, Moritz Egert

TL;DR

The paper investigates boundary-value problems for second-order elliptic operators in divergence form with $t$-independent complex coefficients in the upper half-space and identifies a precise link between interior gradient self-improvement (the Meyers exponent $m_+(\,\mathcal{L}\,)$) and a global boundary-resolvent exponent $p_+(DB)$ from the first-order approach. It develops a two-pronged strategy: globalizing reverse Hölder estimates via a global operator bound and exploiting operator-valued Fourier multipliers through the tangential Fourier transform, connecting $p_+(DB)$ to $m_+(\,\mathcal{L}\,)$. A central result is the equality $p_+(DB)=m_+(\,\mathcal{L}\,)$, with multiple equivalent characterizations through $L_\tau$-Hodge projectors, resolvent estimates, and Weis multiplier theory, unifying interior regularity with boundary $L^p$ solvability. The analysis relies on a careful dimension-raising/lowering framework and a detailed study of the tangential operator family $L_\tau$, culminating in a robust, dimension-independent description of the admissible $p$-ranges for both interior and boundary estimates. This provides a rigorous bridge between Meyers-type interior estimates and global, boundary-focused operator theory for complex-coefficient elliptic systems.

Abstract

The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse H{ö}lder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.

Meyers exponent rules the first-order approach to second-order elliptic boundary value problems

TL;DR

The paper investigates boundary-value problems for second-order elliptic operators in divergence form with -independent complex coefficients in the upper half-space and identifies a precise link between interior gradient self-improvement (the Meyers exponent ) and a global boundary-resolvent exponent from the first-order approach. It develops a two-pronged strategy: globalizing reverse Hölder estimates via a global operator bound and exploiting operator-valued Fourier multipliers through the tangential Fourier transform, connecting to . A central result is the equality , with multiple equivalent characterizations through -Hodge projectors, resolvent estimates, and Weis multiplier theory, unifying interior regularity with boundary solvability. The analysis relies on a careful dimension-raising/lowering framework and a detailed study of the tangential operator family , culminating in a robust, dimension-independent description of the admissible -ranges for both interior and boundary estimates. This provides a rigorous bridge between Meyers-type interior estimates and global, boundary-focused operator theory for complex-coefficient elliptic systems.

Abstract

The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse H{ö}lder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.

Paper Structure

This paper contains 21 sections, 31 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Principal ideas for the proof of Theorem \ref{['Meyers meets DB: Main Thm: Easy formulation']}
  3. Structure of the paper and additional findings
  4. Acknowledgment
  5. Notation
  6. Function spaces and elliptic operators
  7. Tools from $\mathrm{L}^p - \mathrm{L}^q$ off-diagonal theory
  8. $\tau$-dependent Hodge projectors
  9. The first-order Dirac operator $DB$
  10. Global characterization of the Meyers exponent
  11. From $1+n$ to $n$ dimensions
  12. From $n$ to $1+n$ dimensions
  13. Reduction to a resolvent estimate in $1+n$ dimensions
  14. Reduction to an operator-valued multiplier estimate
  15. $\mathrm{L}^2$-results for auxiliary operators
  16. ...and 6 more sections

Key Result

Theorem 1.1

The exponent $p_+(DB)$ from the first-order approach and the Meyers exponent $m_+(\mathcal{L})$ coincide.

Figures (1)

  • Figure 1: Roadmap to Theorem \ref{['Meyers meets DB: Main Thm: Easy formulation']}. The exponent $p_+(DB)$ coincides with the Meyers exponent for $\mathcal{L}$.

Theorems & Definitions (69)

  • Theorem 1.1
  • Definition 3.1
  • Definition 3.3
  • Remark 3.4
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • Definition 5.1
  • Proposition 5.2
  • ...and 59 more