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Double phase quasiconvex functionals and their partial regularity theory

Sunwoo Jeong, Jihoon Ok

Abstract

We consider degenerate nonautonomous energies $$ \int_Ωf(x, Dv)\, dx, $$ for vector-valued functions $v \in W^{1,1}(Ω, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption associated with the double phase function $H(x,t)=t^p + a(x)t^q$. We establish partial Hölder regularity for the gradients of minimizers under suitable, and possibly minimal, regularity assumptions on $H$ and $f$. Our approach relies on two approximation results: $\mathcal{A}$-harmonic approximation and a variational version of the $φ$-harmonic approximation.

Double phase quasiconvex functionals and their partial regularity theory

Abstract

We consider degenerate nonautonomous energies for vector-valued functions , where the integrand satisfies growth and weak uniform quasiconvexity assumption associated with the double phase function . We establish partial Hölder regularity for the gradients of minimizers under suitable, and possibly minimal, regularity assumptions on and . Our approach relies on two approximation results: -harmonic approximation and a variational version of the -harmonic approximation.
Paper Structure (12 sections, 24 theorems, 213 equations)

This paper contains 12 sections, 24 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Basic notation
  5. N-functions
  6. Approximation and regularity results for autonomous problems
  7. Caccioppoli and Self-improving estimates
  8. Self-improving estimates for gradients
  9. Self-improving estimates in the shifted setting
  10. Nondegenerate regime
  11. Degenerate Regime
  12. Iteration

Key Result

Theorem 1.8

Let $H\,:\,\Omega\times[0,\infty)\to[0,\infty)$ be defined as in (H) with (a), and $f:\Omega\times\mathbb{R}^{N\times n}\to\mathbb{R}$ satisfy the assumptions C1--C7. If $u\in W^{1,1}(\Omega,\mathbb{R}^N)$ with $H(\cdot,|Du|)\in L^1(\Omega)$ minimizes $\mathcal{F}$, then there exist $\beta=\beta(n,N Moreover, $\Omega\setminus\Omega_0\subset\Sigma_1\cup\Sigma_2$, where

Theorems & Definitions (35)

  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Proposition 2.4
  • Lemma 2.8
  • Lemma 2.20
  • Lemma 2.21
  • Lemma 2.22
  • Lemma 2.23
  • Lemma 2.27: $\varphi$-minimizing approximation
  • ...and 25 more