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Partial regularity for minimizing constraint maps for the Alt-Phillips energy

Rada Ziganshina

Abstract

In this paper, we establish an $\varepsilon$-regularity theorem for minimizers of an Alt-Phillips type functional subject to constraint maps. We prove that under sufficiently small energy, the minimizers exhibit regularity, and hence proving the smoothness of these maps. From here, we bootstrap to optimal regularity.

Partial regularity for minimizing constraint maps for the Alt-Phillips energy

Abstract

In this paper, we establish an -regularity theorem for minimizers of an Alt-Phillips type functional subject to constraint maps. We prove that under sufficiently small energy, the minimizers exhibit regularity, and hence proving the smoothness of these maps. From here, we bootstrap to optimal regularity.

Paper Structure

This paper contains 5 sections, 11 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Preliminaries and basic tools
  5. Partial regularity

Key Result

Theorem 1.1

Let $u\in W^{1,2}(\Omega;\overline M)$ be a minimizing constraint map for ${\mathcal{E}}_{\lambda,\gamma}$ with $\rho\circ u\leq d$ a.e. in a ball $B_{2R}(x_0)\subset\Omega$. For every $\sigma\in(0,1)$, there are constants $\varepsilon > 0$ small and $c > 1$ large, all depending at most on $n$, $m$, then $u\in C^{1,\alpha}(B_R(x_0);\overline M)$ with $\alpha = \min\{\frac{\gamma}{2-\gamma},\sigma\

Theorems & Definitions (22)

  • Theorem 1.1: e-regularity
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2: Almost monotonicity formula
  • proof
  • Theorem 2.3: Partial regularity due to L
  • Remark 2.4
  • Theorem 2.5
  • proof : Proof of \ref{['eq:camp']}
  • ...and 12 more