On a class of nonlocal obstacle type problems related to the distributional Riesz fractional derivative

In this work, we consider the nonlocal obstacle problem with a given $\psi$ in bounded Lipschitz domain $\Omega$ $\mathbb{R}^{d}$, such that $\mathbb{K}\_\psi^s={v\in H^s\_0(\Omega):v\geq\psi \text{ a.e. }\Omega}\neq\emptyset$, 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 $F$ $H^{-s}(\Omega)$, dual space of fractional Sobolev $H^s\_0(\Omega)$, $0\<s<1$. The operator $\mathcal{L}\_a:H^s\_0(\Omega)\to H^{-s}(\Omega)$ is defined measurable, bounded, strictly positive singular kernel $a(x,y):\mathbb{R}^d\times\mathbb{R}^d\to\[0,\infty)$, 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 (not necessarily symmetric) Dirichlet form, where $\tilde{u},\tilde{v}$ are zero extensions $u$ and $v$ outside respectively. Furthermore, show $\tilde{\mathcal{L}}A=-D^s\cdot AD^s:H^s\_0(\Omega)\to distributional Riesz $D^s$ matrix $A(x)$ corresponds to integral $\mathcal{L}{k\_A}$ well-defined $a=k\_A$. corresponding $s$-fractional $\tilde{\mathcal{L}}\_A$ shown converge as $s\nearrow1$ $H^1\_0(\Omega)$ $-D\cdot AD$ classical gradient $D$. We mainly type problems involving $\mathcal{E}a$ one or two obstacles, well $N$-membranes problem, thereby deriving several results, weak maximum principle, comparison properties, approximation penalization, also Lewy–Stampacchia inequalities. This provides regularity solutions, including global estimate $L^\infty(\Omega)$, local H\\"older solutions when $a$ symmetric, spaces $W^{2s,p}{\mathrm{loc}}(\Omega)$ $C^1(\Omega)$ $\mathcal{L}\_a=(-\Delta)^s$ $s$-Laplacian 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{ }{f}\in \[L^2(\mathbb{R}^d)]^d. These novel results complemented extension inequalities order $H^s\_0(\Omega)$ some remarks on associated $s$-capacity $s$-nonlocal general $\mathcal{L}\_a$.