Higher Sobolev regularity on the mixed local and nonlocal p-Laplace equations
Yuzhou Fang, Dingding Li, Chao Zhang
TL;DR
The paper addresses interior Sobolev regularity for weak solutions to the mixed local and nonlocal p-Laplace equation $-\Delta_p u + A(-\Delta_p)^s u=0$ in a bounded domain. It proves sharp higher-regularity results: for $p\ge2$, $|\nabla u|^{(p-2)/2}\nabla u$ belongs to $W^{1,2}_{\rm loc}$ (and, under suitable bounds, $\nabla u$ attains fractional differentiability $W^{\alpha,q}_{\rm loc}$ with $\alpha<2/q$ for all $q\ge p$); for $1<p\le2$, $u$ gains $W^{2,2}_{\rm loc}$ regularity under local Lipschitz continuity, and in all cases $u\in W^{1+\beta,q}_{\rm loc}$ for any $q\ge\max\{p,2\}$ and suitable $\beta\in(0,2/q)$. The approach combines finite difference quotient techniques, energy estimates, and careful tail analysis to couple the local and nonlocal effects and to obtain uniform estimates across scales. These results extend the high-regularity theory known for purely local or purely nonlocal p-Laplacians to the mixed operator setting, providing new insights into the regularity theory of mixed diffusions with sub- and superquadratic growth.
Abstract
We develop a systematic study of the interior Sobolev regularity of weak solutions to the mixed local and nonlocal $p$-Laplace equations. To be precise, we show that the weak solution $u$ belongs to $W^{2, p}_\mathrm{loc}$ and even $W^{2, 2}_{\rm loc}$ Sobolev spaces in the subquadratic case, while $|\nabla u|^{\frac{p-2}{2}}\nabla u$ is of the class $W^{1, 2}_\mathrm{loc}$ in the superquadratic scenario, both of which coincide with that of the classical $p$-Laplace equations. Moreover, an improved higher fractional differentiability and integrability result $u\in W^{1+β, q}_\mathrm{loc}$ is proved in the full range $p\in (1, \infty)$ for any $q\in [\max\{p, 2\}, \infty)$ and $β\in(0, \frac 2q)$. The main analytical tools are the finite difference quotient technique, suitable energy method and tail estimates. As far as we know, our results are new within the context of such mixed problems.