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Extension Problem and Harnack's Inequality for Some Fractional Operators
The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractionalExpand
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Fractional elliptic equations, Caccioppoli estimates and regularity
Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases}Expand
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Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation
TLDR
We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f( t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. Expand
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Extension problem and fractional operators: semigroups and wave equations
We extend results of Caffarelli–Silvestre and Stinga–Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators ofExpand
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Regularity properties of Schrödinger operators
Abstract Let L be a Schrodinger operator of the form L = − Δ + V , where the nonnegative potential V satisfies a reverse Holder inequality. Using the method of L -harmonic extensions we studyExpand
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Fractional Laplacian on the torus
We study the fractional Laplacian $(-\Delta)^{\sigma/2}$ on the $n$-dimensional torus $\mathbb{T}^n$, $n\geq1$. First, we present a general extension problem that describes \textit{any} fractionalExpand
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Fractional semilinear Neumann problems arising from a fractional Keller–Segel model
We consider the following fractional semilinear Neumann problem on a smooth bounded domain $$\Omega \subset \mathbb {R}^n$$Ω⊂Rn, $$n\ge 2$$n≥2, $$\begin{aligned} {\left\{ \begin{array}{ll}Expand
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Maximum principles, extension problem and inversion for nonlocal one-sided equations
We study one-sided nonlocal equations of the form $$\int_{x_0}^\infty\frac{u(x)-u(x_0)}{(x-x_0)^{1+\alpha}} dx=f(x_0),$$ on the real line. Notice that to compute this nonlocal operator of orderExpand
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User's guide to the fractional Laplacian and the method of semigroups
The \textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularityExpand
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Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications
The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ \[ (-\Delta_h)^su=f, \] for $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0<s<1$, isExpand
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