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 $$ u\in\mathbb{K}\_\psi^s:\quad\langle\mathcal{L}au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s\psi, $$ for $F$ in $H^{-s}(\Omega)$, the dual space of the fractional Sobolev space $H^s\_0(\Omega)$, $0\<s<1$. 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):\mathbb{R}^d\times\mathbb{R}^d\to\[0,\infty)$, by the bilinear form $$ \langle\mathcal{L}au,v\rangle=\mathrm{P.V.}\int{\mathbb{R}^d}\int\_{\mathbb{R}^d} \tilde{v}(x)(\tilde{u}(x)-\tilde{u}(y))a(x,y) ,dy,dx=\mathcal{E}\_a(u,v), $$ 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 $\tilde{\mathcal{L}}A=-D^s\cdot AD^s:H^s\_0(\Omega)\to H^{-s}(\Omega)$ 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 $s\nearrow1$ to the obstacle problem in $H^1\_0(\Omega)$ 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 H\\"older regularity of the solutions when $a$ is symmetric, and local regularity in fractional Sobolev spaces $W^{2s,p}{\mathrm{loc}}(\Omega)$ and in $C^1(\Omega)$ when $\mathcal{L}\_a=(-\Delta)^s$ corresponds to fractional $s$-Laplacian obstacle type problems $$ u\in\mathbb{K}^s\_\psi(\Omega):\quad\int\_{\mathbb{R}^d}(D^su-{f})\cdot D^s(v-u),dx\geq0\quad\forall v\in\mathbb{K}^s\_\psi\text{ for }{f}\in \[L^2(\mathbb{R}^d)]^d. $$ 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 and the $s$-nonlocal obstacle problem for a general $\mathcal{L}\_a$.