Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,γ}$ inclusions

Hongjie Dong, Longjuan Xu

TL;DR

This work analyzes gradient concentration for the nonlinear $p$-Laplacian in a perfect-conductivity model with two closely spaced inclusions. The authors develop flux-based variational techniques and introduce scale functions $\\Theta(\\varepsilon;p)$ and $\\Theta(\\varepsilon;p,\\gamma)$ to capture near-contact behavior, yielding sharp results in two boundary-regularity regimes. For inclusions with partially flat boundaries, the gradient remains uniformly bounded; for $C^{1,\\gamma}$ boundaries, they obtain upper and lower bounds with optimal blow-up rates and precise asymptotics in special cases. The findings illuminate how boundary regularity governs field concentration in nonlinear composites and have potential implications for nonlinear dielectrics and plasticity modeling.

Abstract

In this paper, we investigate the gradient estimates for solutions to the perfect conductivity problem with two closely spaced perfect conductors embedded in a homogeneous matrix, modeled by $p$-Laplacian elliptic equations. We first prove that the gradient of the solution remains bounded when the conductors possess partially ``flat" boundaries. This contrasts with the case involving strictly convex inclusions, where the gradient can blow up. Second, for conductors with $C^{1,γ}$ boundaries ($γ\in(0,1)$), we establish both upper and lower bounds on the gradient, with optimal blow-up rates. Furthermore, we provide precise asymptotic expansions in some special cases.

Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,γ}$ inclusions

TL;DR

This work analyzes gradient concentration for the nonlinear -Laplacian in a perfect-conductivity model with two closely spaced inclusions. The authors develop flux-based variational techniques and introduce scale functions and to capture near-contact behavior, yielding sharp results in two boundary-regularity regimes. For inclusions with partially flat boundaries, the gradient remains uniformly bounded; for boundaries, they obtain upper and lower bounds with optimal blow-up rates and precise asymptotics in special cases. The findings illuminate how boundary regularity governs field concentration in nonlinear composites and have potential implications for nonlinear dielectrics and plasticity modeling.

Abstract

In this paper, we investigate the gradient estimates for solutions to the perfect conductivity problem with two closely spaced perfect conductors embedded in a homogeneous matrix, modeled by -Laplacian elliptic equations. We first prove that the gradient of the solution remains bounded when the conductors possess partially ``flat" boundaries. This contrasts with the case involving strictly convex inclusions, where the gradient can blow up. Second, for conductors with boundaries (), we establish both upper and lower bounds on the gradient, with optimal blow-up rates. Furthermore, we provide precise asymptotic expansions in some special cases.
Paper Structure (8 sections, 12 theorems, 149 equations)

This paper contains 8 sections, 12 theorems, 149 equations.

Table of Contents

  1. Introduction and main results
  2. The case when and have partially "flat" boundaries
  3. The case when and have boundaries
  4. Main ingredients of the proofs
  5. Preliminaries
  6. Proof of Theorem \ref{['thm-flat']}
  7. Proof of Theorem \ref{['thm-gamma']}
  8. Proof of Lemma \ref{['lem-Dv']}

Key Result

Theorem 1.1

Let $h_1$, $h_2$ be two $C^2$ functions satisfying assump-h, $p>1$, $n\geq 2$, and let $u_\varepsilon\in W^{1,p}(\mathcal{D})$ be a weak solution of p-laplace. Then for small $\varepsilon\in(0,100^{-2})$, $|\Sigma'|>0$, and $x\in\Omega_{R}^\varepsilon$, we have where $f_{1}: \mathbb R\rightarrow\mathbb R$ is a function of $\varepsilon$ and ${\bf f}_{1}: \mathbb R^n\times\mathbb R\rightarrow\mathb

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • ...and 15 more