Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global regularity estimates for $p(x)$-Laplacian variational inequalities with singular or degenerate matrix-valued weights

Minh-Phuong Tran, Duc-Quang Bui, Thanh-Nhan Nguyen

TL;DR

The paper addresses global gradient regularity for weak solutions of two-sided obstacle variational inequalities driven by a degenerate-singular $p(x)$-Laplacian with matrix-valued weights. It advances a two-pronged regularity theory, delivering global weighted Calderón-Zygmund-type and weighted Orlicz-type estimates by developing a multilevel comparison scheme that links the original problem to a chain of homogeneous problems with fixed growth and weights. A key innovation is a sharp level-set analysis for fractional maximal operators $_eta$, yielding nearly optimal scaling dependence and enabling global gradient bounds in weighted $L^{ p(dot)}$ spaces and corresponding Orlicz spaces. The results extend Calderón-Zygmund regularity to nonlinear, variable-exponent, degenerate-weight contexts, with potential applications to anisotropic diffusion and related free-boundary phenomena, under minimal geometric and data assumptions.

Abstract

We establish the global gradient bounds for weak solutions to the elliptic variational inequality with two-sided obstructions, associated with a $p(x)$-Laplacian type operator involving degenerate or singular matrix weights. Under the optimal regularity assumptions on the matrix-valued weight, suitable geometric flatness of the domain, and the prescribed data, we aim to investigate the effects of the problem structure on the level of integrability properties of solutions. To this end, we develop regularity in two regards: weighted Calderón-Zygmund-type and general weighted Orlicz-type estimates. A notable feature of our results is that, through a constructive level-set approach, the estimates can be derived with minimal dependence of the scaling parameter on the structural constants. The regularity results are then sharp in the sense that they enable the construction of a level-set estimate with nearly optimal scaling parameters, within admissible parameter sets.

Global regularity estimates for $p(x)$-Laplacian variational inequalities with singular or degenerate matrix-valued weights

TL;DR

The paper addresses global gradient regularity for weak solutions of two-sided obstacle variational inequalities driven by a degenerate-singular -Laplacian with matrix-valued weights. It advances a two-pronged regularity theory, delivering global weighted Calderón-Zygmund-type and weighted Orlicz-type estimates by developing a multilevel comparison scheme that links the original problem to a chain of homogeneous problems with fixed growth and weights. A key innovation is a sharp level-set analysis for fractional maximal operators , yielding nearly optimal scaling dependence and enabling global gradient bounds in weighted spaces and corresponding Orlicz spaces. The results extend Calderón-Zygmund regularity to nonlinear, variable-exponent, degenerate-weight contexts, with potential applications to anisotropic diffusion and related free-boundary phenomena, under minimal geometric and data assumptions.

Abstract

We establish the global gradient bounds for weak solutions to the elliptic variational inequality with two-sided obstructions, associated with a -Laplacian type operator involving degenerate or singular matrix weights. Under the optimal regularity assumptions on the matrix-valued weight, suitable geometric flatness of the domain, and the prescribed data, we aim to investigate the effects of the problem structure on the level of integrability properties of solutions. To this end, we develop regularity in two regards: weighted Calderón-Zygmund-type and general weighted Orlicz-type estimates. A notable feature of our results is that, through a constructive level-set approach, the estimates can be derived with minimal dependence of the scaling parameter on the structural constants. The regularity results are then sharp in the sense that they enable the construction of a level-set estimate with nearly optimal scaling parameters, within admissible parameter sets.
Paper Structure (6 sections, 11 theorems, 189 equations)

This paper contains 6 sections, 11 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Comparison maps
  4. Estimates on level sets
  5. Statements and proofs of main results
  6. Further discussion

Key Result

Proposition 2.14

For given $s \ge 1$, $\beta \in \left[0,\frac{n}{s}\right)$ and $f \in L^{s}(\mathbb{R}^n)$. Then, there exists a constant $C=C(n,s,\beta)>0$ such that for any $\lambda>0$, one has

Theorems & Definitions (27)

  • Definition 2.1: Matrix weight
  • Definition 2.2: Logarithmic mean of weight
  • Remark 2.3
  • Definition 2.4: $\log$-BMO seminorm
  • Remark 2.5
  • Definition 2.6: Variable Lebesgue spaces
  • Definition 2.7: Variable Lebesgue and Sobolev spaces in the intrinsic weighted setting
  • Definition 2.8: Variable Lebesgue and Sobolev spaces in the standard weighted setting
  • Definition 2.9: Muckenhoupt class
  • Definition 2.10: Muckenhoupt class in the variable exponent setting
  • ...and 17 more