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Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

Christopher Irving

TL;DR

The paper studies partial regularity for minimisers of higher-order quasiconvex functionals with natural growth governed by an $N$-function $\varphi$ under the $\Delta_2$ and $\nabla_2$ conditions. It develops an $\varepsilon$-regularity framework by combining a Caccioppoli inequality of the second kind with a harmonic-approximation argument based on a Lipschitz truncation, yielding an excess-decay mechanism that implies $C^{k,\alpha}$ regularity away from a singular set. The results extend known $k=1$ quasiconvex regularity to higher order without requiring quantitative second-derivative bounds on the integrand and, in addition, cover strong local minimisers. This work advances the understanding of regularity under general Orlicz-type growth and higher-order quasiconvexity, with potential implications for vector-valued variational problems in the Orlicz setting.

Abstract

A partial regularity theorem is presented for minimisers of $k$th-order functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an $N$-function satisfying the $Δ_2$ and $\nabla_2$ conditions, assuming no quantitative estimates on the second derivative of the integrand; this is new even in the $k = 1$ case. These results will also be extended to the case of strong local minimisers.

Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

TL;DR

The paper studies partial regularity for minimisers of higher-order quasiconvex functionals with natural growth governed by an -function under the and conditions. It develops an -regularity framework by combining a Caccioppoli inequality of the second kind with a harmonic-approximation argument based on a Lipschitz truncation, yielding an excess-decay mechanism that implies regularity away from a singular set. The results extend known quasiconvex regularity to higher order without requiring quantitative second-derivative bounds on the integrand and, in addition, cover strong local minimisers. This work advances the understanding of regularity under general Orlicz-type growth and higher-order quasiconvexity, with potential implications for vector-valued variational problems in the Orlicz setting.

Abstract

A partial regularity theorem is presented for minimisers of th-order functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an -function satisfying the and conditions, assuming no quantitative estimates on the second derivative of the integrand; this is new even in the case. These results will also be extended to the case of strong local minimisers.
Paper Structure (13 sections, 14 theorems, 122 equations)

This paper contains 13 sections, 14 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Hypotheses and main results
  3. Preliminaries
  4. Notation
  5. $N$-functions
  6. Orlicz-Sobolev spaces
  7. Linear elliptic estimates
  8. A Lipschitz truncation lemma
  9. Proof of the regularity theorem
  10. Caccioppoli inequality of the second kind
  11. Harmonic approximation
  12. Excess decay estimate
  13. Extension to strong local minimisers

Key Result

Theorem 1.2

Let $F$ satisfy Hypotheses hyp:generalgrowth_F and let $M>0,$$\alpha \in (0,1)$ be given. Then there exists $\varepsilon >0$ such that if $u \in \mathop{\mathrm{W}}\nolimits^{k,\varphi}(\Omega,\mathbb R^N)$ minimises (eq:functional_autonomous) where $\Omega \subset \mathbb R^n$ is a bounded domain, then we have $u$ is of class $\mathop{\mathrm{C}}\nolimits^{k,\alpha}$ in $\mathrm{B}_{R/2}(x_0).$

Theorems & Definitions (30)

  • Theorem 1.2: $\varepsilon$-regularity theorem
  • Theorem 1.3: Partial regularity of minimisers
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof : Sketch of proof
  • ...and 20 more