Regularization using different surfaces and the second-order derivatives of 1/r

Abstract

We study the dependence on the surface Σ : g(x) = 1, where g(x) is continuous in ℝ ⁿ ∖{0} and homogeneous of degree 1, of the distributional regularization ℛ_Σ(K(x))∈𝒟′(ℝ ⁿ ),

of a homogeneous function K defined in ℝ ⁿ ∖{0}. We also consider the dependence on Σ of the regularization Pf_Σ(K(x)) given by the Hadamard finite part of the limit when this limit does not exist. We study, in particular, the results on surface dependence for the kernels

the ordinary second-order derivatives in ℝ³, and recover the formulae for the second-order generalized derivatives obtained by Hnizdo [V. Hnizdo, Generalized second-order partial derivatives of 1/r, Eur. J. Phys. 32 (2011), pp. 287–297].

Keywords:

• regularization
• surfaces
• finite part

AMS Subject Classification:

• 46F10