On a Class of Nonlocal Obstacle Type Problems Related to the Distributional Riesz Fractional Derivative

TL;DR

This paper studies nonlocal obstacle problems involving fractional derivatives and general kernels, establishing existence, regularity, and inequalities, and analyzing the limit as the fractional order approaches 1.

Contribution

It introduces new regularity results and inequalities for nonlocal obstacle problems with general kernels, extending classical theory to fractional and non-symmetric cases.

Findings

Established maximum principle and comparison properties.

Derived regularity results including $L^ obreakdash^()$ bounds and local Hf6lder regularity.

Extended Lewy-Stampacchia inequalities to dual spaces.

Abstract

In this work, we consider the nonlocal obstacle problem with a given obstacle $\psi$ in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^{d}$, such that $\mathbb{K}_\psi^s=\{v\in H^s_0(\Omega):v\geq\psi \text{ a.e. in }\Omega\}\neq\emptyset$, given by \[u\in\mathbb{K}_\psi^s:\langle\mathcal{L}_au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_\psi,\] for $F\in H^{-s}(\Omega)$, the dual space of $H^s_0(\Omega)$, $0<s<1$. The nonlocal operator $\mathcal{L}_a:H^s_0(\Omega)\to H^{-s}(\Omega)$ is defined with a measurable, bounded, strictly positive singular kernel $a(x,y)$, possibly not symmetric, by \[\langle\mathcal{L}_au,v\rangle=P.V.\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}v(x)(u(x)-u(y))a(x,y)dydx=\mathcal{E}_a(u,v),\] with $\mathcal{E}_a$ being a Dirichlet form. Also, the fractional operator $\tilde{\mathcal{L}}_A=-D^s\cdot AD^s$ defined with the distributional Riesz $s$-fractional derivative and a bounded matrix $A(x)$ gives a well defined integral singular kernel. The corresponding $s$-fractional obstacle problem converges as $s\nearrow1$ to the obstacle problem in $H^1_0(\Omega)$ with the operator $-D\cdot AD$ given with the gradient $D$. We mainly consider problems involving the bilinear form $\mathcal{E}_a$ with one or two obstacles, and the N-membranes problem, deriving a weak maximum principle, comparison properties, approximation by bounded penalization, and the Lewy-Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in $L^\infty(\Omega)$, local H\"older regularity when $a$ is symmetric, and local regularity in $W^{2s,p}_{loc}(\Omega)$ and $C^1(\Omega)$ for fractional $s$-Laplacian obstacle-type problems. These novel results are complemented with the extension of the Lewy-Stampacchia inequalities to the order dual of $H^s_0(\Omega)$ and some remarks on the associated $s$-capacity for general $\mathcal{L}_a$.