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 ${\mathbb{K}}_{\psi }^s=\{v\in H_0^s(\Omega ):v\ge \psi \text{ a.e. in }\Omega \}\ne \varnothing$, given by \[u\in\mathbb{K}_ψ^s:\langle\mathcal{L}_au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_ψ,\] for $F\in H^{-s}(\Omega )$, the dual space of $H_0^s(\Omega )$, $0<s<1$. The nonlocal operator ${\mathcal{L}}_a:H_0^s(\Omega )\to H^{-s}(\Omega )$ 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 ${\tilde{\mathcal{L}}}_A=-D^s\cdot A{D}^s$ 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 $s\nearrow 1$ to the obstacle problem in $H_0^1(\Omega )$ 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 Hölder regularity when $a$ is symmetric, and local regularity in $W_{loc}^{2s,p}(\Omega )$ 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 $H_0^s(\Omega )$ and some remarks on the associated $s$-capacity for general ${\mathcal{L}}_a$.

Cited by in corpus (6)

• Regularity estimates for fractional orthotropic $p$-Laplacians of mixed order
• On the Obstacle Problem in Fractional Generalised Orlicz Spaces
• Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems
• Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form
• On the Stability of the $s$-Nonlocal $p$-Obstacle Problem and their Coincidence Sets and Free Boundaries
• Nonlocal Lagrange multipliers and transport densities