On the weak Harnack inequalities for nonlocal double phase problems

TL;DR

This paper establishes weak Harnack inequalities for nonlocal double phase functionals, advancing understanding of their regularity properties through measure estimates and boundedness results.

Contribution

It introduces new measure theoretical estimates for nonlocal double phase problems and addresses the interaction between coefficients and growth exponents.

Findings

Proved weak Harnack inequalities for nonlocal double phase functionals.

Developed measure estimates based on nonlocal Caccioppoli inequalities.

Provided boundedness results for minimizers of these functionals.

Abstract

This paper is devoted to studying the weak Harnack inequalities for nonlocal double phase functionals by using expansion of positivity, whose prototype is $$ \iint_{\mathbb{R}^n\times\mathbb{R}^n} \left(\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}+a(x,y)\frac{|u(x)-u(y)|^q}{|x-y|^{n+tq}}\right) \,dxdy $$ with $a\ge0$ and $0<s\le t<1<p\le q$. The core of our approach is to establish several measure theoretical estimates based on the nonlocal Caccioppoli-type inequality, where the challenges consist in controlling subtle interaction between the pointwise behaviour of modulating coefficient and the growth exponents. Meanwhile, a quantitative boundedness result on the minimizer of such functionals is also discussed.