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

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 $\mathcal{L}_a$.