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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 fractional
Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation
TLDR
The regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator, and a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ and parabolic maximum principles are developed.
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 $$\begin{cases}
Extension problem and fractional operators: semigroups and wave equations
We extend results of Caffarelli–Silvestre and Stinga–Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of
Maximum principles, extension problem and inversion for nonlocal one-sided equations
We study one-sided nonlocal equations of the form $$\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 order
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 $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0<s<1$, is
User’s guide to the fractional Laplacian and the method of semigroups
  • P. Stinga
  • Mathematics
    Fractional Differential Equations
  • 15 August 2018
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 regularity
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, $$\begin{aligned} {\left\{ \begin{array}{ll}
Regularity estimates in Hölder spaces for Schrödinger operators via a $$T1$$ theorem
We derive Hölder regularity estimates for operators associated with a time-independent Schrödinger operator of the form $$-\Delta +V$$. The results are obtained by checking a certain condition on the
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} fractional
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