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The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

Simone Ciani, Eurica Henriques, Igor i. Skrypnik

Abstract

In this work we prove that the non-negative functions $u \in L^s_{loc}(Ω)$, for some $s>0$, belonging to the De Giorgi classes \begin{equation}\label{eq0.1} \fint\limits_{B_{r(1-σ)}(x_{0})} \big|\nabla \big(u-k\big)_{-}\big|^{p}\, dx \leqslant \frac{c}{σ^{q}} \,Λ\big(x_{0}, r, k\big)\bigg(\frac{k}{r}\bigg)^{p}\bigg(\frac{\big|B_{r}(x_{0})\cap\big\{u\leqslant k\big\}\big|}{|B_{r}(x_{0})|}\bigg)^{1-δ}, \end{equation} under proper assumptions on $Λ$, satisfy a weak Harnack inequality with a constant depending on the $L^s$-norm of $u$. Under suitable assumptions on $Λ$, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying \eqref{eq0.1}; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents.

The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

Abstract

In this work we prove that the non-negative functions , for some , belonging to the De Giorgi classes \begin{equation}\label{eq0.1} \fint\limits_{B_{r(1-σ)}(x_{0})} \big|\nabla \big(u-k\big)_{-}\big|^{p}\, dx \leqslant \frac{c}{σ^{q}} \,Λ\big(x_{0}, r, k\big)\bigg(\frac{k}{r}\bigg)^{p}\bigg(\frac{\big|B_{r}(x_{0})\cap\big\{u\leqslant k\big\}\big|}{|B_{r}(x_{0})|}\bigg)^{1-δ}, \end{equation} under proper assumptions on , satisfy a weak Harnack inequality with a constant depending on the -norm of . Under suitable assumptions on , the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying \eqref{eq0.1}; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents.
Paper Structure (8 sections, 6 theorems, 79 equations)

This paper contains 8 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction and main results
  2. Auxiliary Results
  3. The Local Clustering Lemma
  4. Measurable theoretical results on the expansion of positivity
  5. The Weak Harnack Inequality: proof of Theorem \ref{['th1.1']}
  6. Expansion of Positivity
  7. Proof of Theorem \ref{['th1.1']}
  8. Applications of Theorem \ref{['th1.1']}

Key Result

Theorem 1.1

Let $s\ge 0$ be a fixed number, and let $\Lambda: \Omega \times \mathbb{R}^+ \times \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function satisfying the properties $(\Lambda_s)$-$(\lambda)$ above. If $u$ is a non-negative member of $DG^{-}_{p, \Lambda}(\Omega)\cap L^{s}_{loc}(\Omega)$, then there exi provided that $B_{8\rho}(x_{0})\subset \Omega$.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • proof
  • ...and 5 more