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Asymptotically quasiconvex functionals with general growth conditions

Francesca Angrisani

TL;DR

This paper tackles local regularity for local minimizers of autonomous vectorial integrals with general Orlicz-type growth under asymptotic quasiconvexity at infinity. The authors develop a framework combining a Caccioppoli inequality, an $\mathcal{A}$-harmonic approximation, and an excess-decay argument in Orlicz spaces to establish partial $C^{1,\alpha}$ regularity at points where the gradient is large and Lipschitz regularity on a dense open subset, for all pairs of Young functions $(\varphi,\psi)$ satisfying the $\Delta_2$ condition. The approach extends existing results in the scalar and vectorial cases from homogeneous power-growth to general, non-homogeneous growth, covering both subquadratic and superquadratic regimes. The results provide optimal partial regularity in this broad setting and supply precise machinery (excess decay and $\mathcal{A}$-harmonic approximation) applicable to a wide class of variational problems with Orlicz growth.

Abstract

We establish local regularity results for minimizers of autonomous vectorial integrals of Calculus of Variations, assuming $ψ$-growth conditions and imposing $\varphi$-quasiconvexity only in an asymptotic sense, both in the sub-quadratic and super-quadratic case. In particular, we obtain $C^{1,α}$ regularity at points $x_0$ where $Du$ is sufficiently large in a neighborhood of $x_0$, as well as Lipschitz regularity on a dense set. \ The results hold for all pairs of Young functions $(\varphi, ψ)$ satisfying the $Δ_2$ condition.

Asymptotically quasiconvex functionals with general growth conditions

TL;DR

This paper tackles local regularity for local minimizers of autonomous vectorial integrals with general Orlicz-type growth under asymptotic quasiconvexity at infinity. The authors develop a framework combining a Caccioppoli inequality, an -harmonic approximation, and an excess-decay argument in Orlicz spaces to establish partial regularity at points where the gradient is large and Lipschitz regularity on a dense open subset, for all pairs of Young functions satisfying the condition. The approach extends existing results in the scalar and vectorial cases from homogeneous power-growth to general, non-homogeneous growth, covering both subquadratic and superquadratic regimes. The results provide optimal partial regularity in this broad setting and supply precise machinery (excess decay and -harmonic approximation) applicable to a wide class of variational problems with Orlicz growth.

Abstract

We establish local regularity results for minimizers of autonomous vectorial integrals of Calculus of Variations, assuming -growth conditions and imposing -quasiconvexity only in an asymptotic sense, both in the sub-quadratic and super-quadratic case. In particular, we obtain regularity at points where is sufficiently large in a neighborhood of , as well as Lipschitz regularity on a dense set. \ The results hold for all pairs of Young functions satisfying the condition.
Paper Structure (9 sections, 14 theorems, 105 equations)

This paper contains 9 sections, 14 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Young Functions and Their Properties
  3. Examples
  4. Technical Lemmas and Definitions
  5. Outline of the Proof of the Main Theorem and Preliminary Lemmas
  6. Caccioppoli Inequality
  7. $\mathcal{A}$-harmonicity
  8. Excess Decay Estimate
  9. Acknowledgements

Key Result

Theorem 1

Let $f,\varphi,\psi$ satisfy hypotheses $(H.0)$, $(H.1)$, $(H.2)$, $(H.3)$, and $(H.4)$, and let $u$ be a local minimizer of the corresponding functional $\mathcal{F}$. Suppose that $z_0\in \mathbb{R}^{nN}$ satisfies $|z_0|>M+1$ and that there exists $x_0 \in \mathbb{R}^n$ such that then $x_0 \in \mathop{\mathrm{\text{Reg}}}\limits(u)$, where $\mathop{\mathrm{\text{Reg}}}\limits(u)=\{x \in \Omega

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 2
  • Definition
  • Definition
  • Definition
  • Lemma 3
  • Lemma 4
  • Definition : Excess
  • Remark 1
  • Lemma 5
  • ...and 18 more