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Partial regularity for variational integrals with Morrey-Hölder zero-order terms, and the limit exponent in Massari's regularity theorem

Thomas Schmidt, Jule Helena Schütt

Abstract

We revisit the partial $\mathrm{C}^{1,α}$ regularity theory for minimizers of non-parametric integrals with emphasis on sharp dependence of the Hölder exponent $α$ on structural assumptions for general zero-order terms. A particular case of our conclusions carries over to the parametric setting of Massari's regularity theorem for prescribed-mean-curvature hypersurfaces and there confirms optimal regularity up to the limit exponent.

Partial regularity for variational integrals with Morrey-Hölder zero-order terms, and the limit exponent in Massari's regularity theorem

Abstract

We revisit the partial regularity theory for minimizers of non-parametric integrals with emphasis on sharp dependence of the Hölder exponent on structural assumptions for general zero-order terms. A particular case of our conclusions carries over to the parametric setting of Massari's regularity theorem for prescribed-mean-curvature hypersurfaces and there confirms optimal regularity up to the limit exponent.
Paper Structure (12 sections, 21 theorems, 123 equations, 1 figure)

This paper contains 12 sections, 21 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overall assumptions and settings
  4. Caccioppoli inequality
  5. Approximate A-harmonicity
  6. Excess estimates
  7. Proofs of the partial regularity theorems
  8. Basic regularity conclusion
  9. The refined Hölder exponent and a priori L infinity minimizers
  10. Non-uniform ellipticity and a priori W{1,\\infty} minimizers
  11. Lp-Hölder zero-order terms and Lp-W{1,r} zero-order terms
  12. The optimal Hölder exponent in Massari's regularity theorem

Key Result

Theorem 1.2

We consider a $2$-strictly quasiconvex (in the sense of the later Definition DEF: quasiconvex)$\mathrm{C}^2$ integrand $f\colon\mathds{R}^{N\times n}\to\mathds{R}$ which has at most quadratic growth in the sense of $\limsup_{|z|\to\infty}\frac{|f(z)|}{|z|^2}<\infty$. Moreover, in case $n\ge3$, we ab and with a non-negative function $\Gamma$ which in turn satisfies, for the exponents the two Morre

Figures (1)

  • Figure 1: Local parameterization of $E$ via $u$ and of test sets $F$ via $w$

Theorems & Definitions (46)

  • Definition 1.1: local minimizers
  • Theorem 1.2: partial regularity for variational integrals with Morrey-Hölder zero-order integrand
  • Theorem 1.3: partial regularity for a priori $\mathrm{L}^\infty_\mathrm{loc}$ minimizers
  • Theorem 1.4: partial regularity for a priori $\mathrm{W}^{1,\infty}_\mathrm{loc}$ minimizers in non-uniformly elliptic cases
  • Theorem 1.5: optimal Massari-type regularity
  • Definition 2.1: Morrey spaces
  • Remark 2.2
  • Definition 2.3: ($2$-strict) quasiconvexity
  • Remark 2.4
  • Lemma 2.5
  • ...and 36 more