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Universal Potential Estimates for Mixed Local and Nonlocal Nonlinear Measure Data Problems

Lingwei Ma, Qi Xiong, Zhenqiu Zhang

TL;DR

The paper develops a comprehensive nonlinear potential theory for mixed local and nonlocal nonlinear elliptic equations with measure data, establishing universal pointwise estimates for solutions and their gradients in terms of Wolff and Riesz potentials and fractional maximal functions. By introducing a novel fractional maximal framework, the authors capture both local and nonlocal interactions under minimal coefficient regularity and measurable kernels, proving oscillation and Holder-type estimates valid in sub- and superquadratic regimes. The results extend and unify prior local and nonlocal potential theories, provide robust tail-controlled bounds, and accommodate SOLA solutions, thereby offering a powerful toolkit for Calderón–Zygmund-type regularity in mixed diffusion problems. This advances the understanding of regularity for mixed diffusion models and has potential implications for applications where both local and nonlocal effects interact with measure data.

Abstract

This paper presents the nonlinear potential theory for mixed local and nonlocal $p$-Laplace type equations with coefficients and measure data, involving both superquadratic and subquadratic cases. We prove a class of universal pointwise estimates for the solution and its gradient via Riesz and Wolff potentials. These are achieved by imposing various low regularity conditions on the coefficient of the local term, while the kernel coefficient for the nonlocal term is merely assumed to be measurable. The key to these proofs lies in introducing a novel fractional maximum function that can capture both local and nonlocal features simultaneously, and in establishing pointwise estimates for such maximum operators of the solution and its gradient. Notably, our universal potential estimates not only precisely characterize the oscillations of solutions, but also identify the borderline case that bounds their size, thereby refining the pointwise potential estimates available in earlier work.

Universal Potential Estimates for Mixed Local and Nonlocal Nonlinear Measure Data Problems

TL;DR

The paper develops a comprehensive nonlinear potential theory for mixed local and nonlocal nonlinear elliptic equations with measure data, establishing universal pointwise estimates for solutions and their gradients in terms of Wolff and Riesz potentials and fractional maximal functions. By introducing a novel fractional maximal framework, the authors capture both local and nonlocal interactions under minimal coefficient regularity and measurable kernels, proving oscillation and Holder-type estimates valid in sub- and superquadratic regimes. The results extend and unify prior local and nonlocal potential theories, provide robust tail-controlled bounds, and accommodate SOLA solutions, thereby offering a powerful toolkit for Calderón–Zygmund-type regularity in mixed diffusion problems. This advances the understanding of regularity for mixed diffusion models and has potential implications for applications where both local and nonlocal effects interact with measure data.

Abstract

This paper presents the nonlinear potential theory for mixed local and nonlocal -Laplace type equations with coefficients and measure data, involving both superquadratic and subquadratic cases. We prove a class of universal pointwise estimates for the solution and its gradient via Riesz and Wolff potentials. These are achieved by imposing various low regularity conditions on the coefficient of the local term, while the kernel coefficient for the nonlocal term is merely assumed to be measurable. The key to these proofs lies in introducing a novel fractional maximum function that can capture both local and nonlocal features simultaneously, and in establishing pointwise estimates for such maximum operators of the solution and its gradient. Notably, our universal potential estimates not only precisely characterize the oscillations of solutions, but also identify the borderline case that bounds their size, thereby refining the pointwise potential estimates available in earlier work.
Paper Structure (11 sections, 28 theorems, 160 equations)

This paper contains 11 sections, 28 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Auxiliary results
  4. Regularity for homogeneous equations
  5. Comparison estimates and applications
  6. Comparison between $Dv$ and $Dw$
  7. Comparison between $Du$ and $Dv$
  8. Pointwise estimates of nonlocal fractional maximal functions
  9. Universal potential estimates
  10. Oscillation estimates for solutions
  11. Oscillation estimates for gradient

Key Result

Theorem 1.2

Let $u\in W^{1,p}_{\rm loc}(\Omega)\cap {\mathcal{L}}^{p-1}_{sp}(\mathbb{R}^n)$ be a weak solution to eq1 under assumptions growth1--ellip with $s \in (0,1)$, $p>2-\frac{1}{n}$ and $q_0=\max\{1,p-1\}$. Given $B_R(x_0)\subset\Omega$ with $R\leq R_0$ for some $R_0=R_0(n,p,s,\Lambda,\operatorname{diam} holds uniformly in $\alpha\in[0,\tilde{\alpha}]$, where $\alpha_m$ is given by Lemma v-holder.

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 37 more