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Partial regularity for degenerate systems of double phase type

Jihoon Ok, Giovanni Scilla, Bianca Stroffolini

Abstract

We study partial regularity for degenerate elliptic systems of double-phase type, where the growth function is given by $H(x,t)=t^p+a(x)t^q$ with $1<p\leq q$ and $a(x)$ a nonnegative $C^{0,α}$-continuous function. Our main result proves that if $\frac{q}{p}\leq 1+\fracα{n}$, the gradient of any weak solution is locally Hölder continuous, except on a set of measure zero.

Partial regularity for degenerate systems of double phase type

Abstract

We study partial regularity for degenerate elliptic systems of double-phase type, where the growth function is given by with and a nonnegative -continuous function. Our main result proves that if , the gradient of any weak solution is locally Hölder continuous, except on a set of measure zero.

Paper Structure

This paper contains 15 sections, 21 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries and auxiliary results
  4. Basic notation
  5. Some basic facts on $N$--functions
  6. $\mathcal{A}$-harmonic and $\varphi$-harmonic functions
  7. Harmonic type approximation results
  8. Caccioppoli and reverse Hölder type estimates
  9. Caccioppoli type estimates
  10. Reverse Hölder type estimates
  11. Decay estimates for excess functionals
  12. Non degenerate regime: almost $\mathcal{A}$-harmonic functions
  13. Degenerate regime: almost $\varphi$-harmonic functions
  14. Iteration in the nondegenerate regime
  15. Proof of Theorem \ref{['thm:1:main-thm']}

Key Result

Theorem 1.1

Let $H:\Omega\times[0,\infty)\to[0,\infty)$ be defined as in eq:H complying with eq:pq, and ${\bf A}:\Omega\times \mathbb{R}^{N\times n}\to \mathbb{R}^{N\times n}$ comply with eq:1.8ok1--eq:degenereassump. If $\bfu \in W^{1,1}(\Omega;\mathbb{R}^N)$ with $H(\cdot,|D\bfu|)\in L^1(\Omega)$ is a weak so Moreover, $\Omega \setminus \Omega_0\subset \Sigma_1\cup\Sigma_2$ where where $H^-_{B_r(x_0)}(t)$

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3: Sobolev-Poincaré inequality
  • Lemma 2.4: Sobolev-Poincaré inequality
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 2.7
  • ...and 24 more