Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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 $\mathcal{L}_a$.

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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: supr-con