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

for $F$ in $H^{-s}(\Omega )$, the dual space of the fractional Sobolev space $H_0^s(\Omega )$, $0<s<1$. The nonlocal operator ${\mathcal{L}}_a:H_0^s(\Omega )\to H^{-s}(\Omega )$ is defined with a measurable, bounded, strictly positive singular kernel $a(x,y):{\mathbb{R}}^d\times {\mathbb{R}}^d\to [0,\infty )$, by the bilinear form

which is a (not necessarily symmetric) Dirichlet form, where $\tilde{u},\tilde{v}$ are the zero extensions of $u$ and $v$ outside $\Omega$ respectively. Furthermore, we show that the fractional operator ${\tilde{\mathcal{L}}}_A=-D^s\cdot A{D}^s:H_0^s(\Omega )\to H^{-s}(\Omega )$ defined with the distributional Riesz fractional $D^s$ and with a measurable, bounded matrix $A(x)$ corresponds to a nonlocal integral operator ${\mathcal{L}}_{k_A}$ with a well-defined integral singular kernel $a=k_A$. The corresponding $s$-fractional obstacle problem for ${\tilde{\mathcal{L}}}_A$ is shown to converge as $s\nearrow 1$ to the obstacle problem in $H_0^1(\Omega )$ with the operator $-D\cdot AD$ given with the classical gradient $D$.

We mainly consider obstacle type problems involving the bilinear form ${\mathcal{E}}_a$ with one or two obstacles, as well as the $N$-membranes problem, thereby deriving several results, such as the weak maximum principle, comparison properties, approximation by bounded penalization, and also the Lewy–Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in $L^{\infty }(\Omega )$, local Hölder regularity of the solutions when $a$ is symmetric, and local regularity in fractional Sobolev spaces $W_{\mathrm{loc}}^{2s,p}(\Omega )$ and in $C^1(\Omega )$ when ${\mathcal{L}}_a=(-\Delta {)}^s$ corresponds to 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 and the $s$-nonlocal obstacle problem for a general ${\mathcal{L}}_a$.