P-adic Laplacian in local fields. (July 2016)

• Record Type: Journal Article
• Authors: Li, Yin; Qiu, Hua
• Is Part Of: Nonlinear analysis. Volume 139(2016)
• Page Start: 131
• Page End: 151
• Publication Date: 2016-07
• Subjects: 11F85 -- 46S10 -- 47G30; p-adic fields -- Laplacian -- Pseudo-differential operators -- Eigen-values -- Cauchy problem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.02.025
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper a family of multi-dimensional fractional differential operators T α and their corresponding pseudo-differential equations over p -adic fields are investigated. The test function class D ( Q p n ) and distribution class D ′ ( Q p n ) are invariant under the actions of these operators. The p -adic Laplacian Δ p and a fundamental solution of the Laplace equation are constructed. We study the spectral properties of the Laplacian Δ p, and obtain an orthonormal basis of the eigen-functions of this operator in L 2 ( Q p n ) . Furthermore, the Cauchy problems for the wave and heat equations on the p -adic fields related to Δ p are also studied.