Inverse problems for the fractional-Laplacian with lower order non-local perturbations

2021 ◽  
Vol 374 (5) ◽  
pp. 3053-3075
Author(s):  
S. Bhattacharyya ◽  
T. Ghosh ◽  
G. Uhlmann
Keyword(s):  
Inverse Problems ◽  
Lower Order ◽  
Fractional Laplacian ◽  
Local Perturbations ◽  
Non Local

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Humberto Prado ◽  
Margarita Rivero ◽  
Juan J. Trujillo ◽  
M. Pilar Velasco
Keyword(s):  
Fractional Integral ◽  
Fractional Laplacian ◽  
Singular Integrals ◽  
Differential Operators ◽  
Riesz Potential ◽  
Fractional Power ◽  
Potential Operators ◽  
Relevant Role ◽  
Non Local ◽  
Fractional Power Of Operators

AbstractThe non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

scholarly journals Non-local regularization of inverse problems

2011 ◽  
Vol 5 (2) ◽  
pp. 511-530 ◽  
Author(s):  
Gabriel Peyré ◽  
◽  
Sébastien Bougleux ◽  
Laurent Cohen ◽  
Keyword(s):  
Inverse Problems ◽  
Local Regularization ◽  
Non Local

scholarly journals Some Remarks on the Duality Method for Integro-Differential Equations with Measure Data

2016 ◽  
Vol 16 (1) ◽  
pp. 115-124 ◽  
Author(s):  
Francesco Petitta
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Fractional Order ◽  
Radon Measure ◽  
Fractional Laplacian ◽  
Measure Data ◽  
Duality Method ◽  
Non Local ◽  
The One

AbstractWe deal with existence, uniqueness and regularity for solutions of the boundary value problem$\left\{\begin{aligned} \displaystyle\mathcal{L}^{s}u&\displaystyle=\mu&&% \displaystyle\text{in }\Omega,\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\mathbb{R}^{n}% \backslash\Omega,\end{aligned}\right.$where Ω is a bounded domain of ${\mathbb{R}^{n}}$, μ is a bounded Radon measure on Ω, and ${\mathcal{L}^{s}}$ is a non-local operator of fractional order s whose kernel K is comparable with the one of the fractional Laplacian.

scholarly journals On the spectrum of two different fractional operators

2014 ◽  
Vol 144 (4) ◽  
pp. 831-855 ◽  
Author(s):  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Fractional Operators ◽  
Local Operators ◽  
Non Local ◽  
Definition Of ◽  
Image Position ◽  
The Laplace Operator ◽  
Fractional Type

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

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