An Example of the Generalized Fractional Laplacian on Rn

2020 ◽  
Author(s):  
Chenkuan Li
Keyword(s):  
Unit Sphere ◽  
Fractional Laplacian ◽  
Special Functions ◽  
Multidimensional Surface ◽  
Surface Integrals

Using the normalization of the fractional Laplacian (-△)su(x) over the space Ck(Rn ) for all s > 0 and s ≠ 1, 2, ···, we evaluate (-△)se-||x||2 based on the multidimensional surface integrals over the unit sphere and special functions.

On the generalized fractional Laplacian

2021 ◽  
Vol 24 (6) ◽  
pp. 1797-1830
Author(s):  
Chenkuan Li
Keyword(s):  
Fractional Laplacian ◽  
Special Functions ◽  
Computational Techniques ◽  
Residue Theorem ◽  
End Points ◽  
Proper Subspace ◽  
Classical Sense ◽  
First Time ◽  
Cauchy’S Residue Theorem ◽  
Integral Identities

Abstract The objective of this paper is, for the first time, to extend the fractional Laplacian (−△) s u(x) over the space Ck (Rn ) (which contains S(Rn ) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in Rn . We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.

scholarly journals Numerical integration on the sphere

1982 ◽  
Vol 23 (3) ◽  
pp. 332-347 ◽  
Author(s):  
Kendall Atkinson
Keyword(s):  
Finite Element ◽  
Numerical Integration ◽  
Potential Theory ◽  
Double Layer ◽  
Unit Sphere ◽  
Gaussian Quadrature ◽  
Integration Methods ◽  
Numerical Integration Methods ◽  
Element Type ◽  
Surface Integrals

AbstractThis is a discussion of some numerical integration methods for surface integrals over the unit sphere in R3. Product Gaussian quadrature and finite-element type methods are considered. The paper concludes with a discussion of the evaluation of singular double layer integrals arising in potential theory.

scholarly journals On the Generalized Riesz Derivative

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1089
Author(s):  
Chenkuan Li ◽  
Joshua Beaudin
Keyword(s):  
Banach Space ◽  
Integral Representation ◽  
Unit Sphere ◽  
Normed Space ◽  
Continuous Mapping ◽  
End Points ◽  
Riesz Derivative ◽  
One To One ◽  
Surface Integrals

The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

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