Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no Hölder condition
Yong-Cheol Kim
TL;DR
This work extends nonlocal Harnack theory to nonlocal double phase equations with a bounded modulating coefficient that may lack Hölder continuity. It adapts the De Giorgi–Nash–Moser method to the operator ${\mathcal L}$, establishing a nonlocal Caccioppoli inequality, a logarithmic BMO estimate, and tail-controlled iteration to derive nonlocal weak Harnack and Harnack inequalities under the regime ${qt}\le ps\le n$ and ${1<p\le q\le p_s^*}$. Key contributions include explicit local boundedness, weak Harnack for supersolutions, and the first nonlocal Harnack inequalities for weak solutions without Hölder regularity on the modulating coefficient. The results provide a robust framework for nonlocal double-phase problems and pave the way for future work on the complementary regime ${ps}\le qt$ and related nonlocal growth conditions.
Abstract
In this paper, by applying the De Giorgi-Nash-Moser theory we prove nonlocal Harnack inequalities for (locally nonnegative in $Ω$) weak solutions to nolocal double phase equations \begin{equation*}\begin{cases}\cL u =0 & \text{ in $Ω$,} \\ u=g & \text{ in $\BR^n\sΩ$ } \end{cases}\end{equation*} where $Ω\subset\BR^n$ ($n\ge 2$) is a bounded domain with Lipschitz boundary, $\cL$ is the nonlocal double phase operator $\cL$ given by \begin{equation*}\begin{split}\cL u(x)=&\pv\int_{\BR^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{ps}(x,y)\,dy \\ &+\pv\int_{\BR^n}\fa(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{qt}(x,y)\,dy, \end{split} \end{equation*} $0<\fa(x,y) = \fa(y,x) \le \|\fa\|_{L^\iy(\BR^n\times\BR^n)} < \iy$ and $ps\ge qt$ for $0<s,t<1<p\le q<\iy$. In addition, we get local boundedness with explicit formula and weak Harnack inequalities for their weak supersolutions.