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Calderón-Zygmund gradient estimates for $p$-Laplace systems with BMO complex coefficients

Van-Chuong Quach, Thanh-Nhan Nguyen, Minh-Phuong Tran

TL;DR

The paper addresses global gradient bounds for divergence-form degenerate elliptic systems with complex coefficients, allowing leading coefficients that are only small in BMO and considered on Reifenberg-flat domains. It develops a Calderón-Zygmund-type framework using the fractional maximal operators $\mathbf{M}_{\beta}$ and a level-set decay approach driven by local comparison with homogeneous problems, aided by a real-variable reformulation via $\mathcal{Q}_1$ and $\mathcal{Q}_2$. The main contributions are quantitative $L^q$ gradient estimates and Morrey-space regularity for weak solutions, together with existence and uniqueness, extending regularity theory in the complex-valued setting beyond previous results such as KV25. The results provide robust global gradient control for complex $p$-Laplacian-type systems, with potential applications to complex media and inverse problems where non-real coefficients arise.

Abstract

This work is concerned with global gradient bounds for a class of divergence-form degenerate elliptic systems with complex-valued coefficients. Notably, the leading coefficients are merely required to be sufficiently small in BMO, which is strictly weaker than the VMO condition. In the complex setting, the well-posedness of this problem was recently investigated in [W. Kim, M. Vestberg, Existence, uniqueness and regularity for elliptic $p$-Laplace systems with complex coefficients,arXiv:2503.18932], where the authors established a strong accretivity condition on the leading coefficients, and this structural condition allows them to derive Schauder-type estimates for weak solutions. In our study, it has already been observed that gaining existence and uniqueness of weak solutions is possible under a natural and less restrictive assumption on the complex-valued coefficients. Following this direction, we prove a global Caderón-Zygmund-type estimate for weak solutions, from which the Morrey-space regularity follows as a consequence. This paper is a contribution to the better understanding of solution behavior and may be viewed as part of a series of works aimed at extending regularity theory in the complex-valued setting.

Calderón-Zygmund gradient estimates for $p$-Laplace systems with BMO complex coefficients

TL;DR

The paper addresses global gradient bounds for divergence-form degenerate elliptic systems with complex coefficients, allowing leading coefficients that are only small in BMO and considered on Reifenberg-flat domains. It develops a Calderón-Zygmund-type framework using the fractional maximal operators and a level-set decay approach driven by local comparison with homogeneous problems, aided by a real-variable reformulation via and . The main contributions are quantitative gradient estimates and Morrey-space regularity for weak solutions, together with existence and uniqueness, extending regularity theory in the complex-valued setting beyond previous results such as KV25. The results provide robust global gradient control for complex -Laplacian-type systems, with potential applications to complex media and inverse problems where non-real coefficients arise.

Abstract

This work is concerned with global gradient bounds for a class of divergence-form degenerate elliptic systems with complex-valued coefficients. Notably, the leading coefficients are merely required to be sufficiently small in BMO, which is strictly weaker than the VMO condition. In the complex setting, the well-posedness of this problem was recently investigated in [W. Kim, M. Vestberg, Existence, uniqueness and regularity for elliptic -Laplace systems with complex coefficients,arXiv:2503.18932], where the authors established a strong accretivity condition on the leading coefficients, and this structural condition allows them to derive Schauder-type estimates for weak solutions. In our study, it has already been observed that gaining existence and uniqueness of weak solutions is possible under a natural and less restrictive assumption on the complex-valued coefficients. Following this direction, we prove a global Caderón-Zygmund-type estimate for weak solutions, from which the Morrey-space regularity follows as a consequence. This paper is a contribution to the better understanding of solution behavior and may be viewed as part of a series of works aimed at extending regularity theory in the complex-valued setting.
Paper Structure (5 sections, 8 theorems, 165 equations)

This paper contains 5 sections, 8 theorems, 165 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and function spaces
  3. Existence and uniqueness
  4. Homogeneous problems and comparison maps
  5. Proofs of the main results

Key Result

Theorem 1.3

Let $p>1$, $n \ge 2$, $N \ge 1$ and $\mu \in [0,1]$. Suppose that the right-hand side source term $\mathbf{F} \in L^p(\Omega;\mathbb{C}^{N \times n})$ and complex-valued coefficient $\mathfrak{a}$ satisfies assumption (A1). Then, the system eq-main admits a unique weak solution $\mathbf{u} \in W_0^{

Theorems & Definitions (11)

  • Definition 1.1: Weak solution
  • Definition 1.2: Fractional maximal operators
  • Theorem 1.3: Existence and uniqueness
  • Theorem 1.4: Calderón-Zygmund estimate
  • Theorem 1.5: Morrey regularity
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Remark 3.2
  • Lemma 4.1: Local comparison maps
  • ...and 1 more