Fundamental solutions for the Tricomi operator
type: Item
instance of: scholarly article
title: Fundamental solutions for the Tricomi operator
zbMATH Open document ID: 0945.35063
DOI: 10.1215/S0012-7094-99-09814-9
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The main aim of the paper is to determine fundamental solutions for the Tricomi equation \({\mathfrak I}u= yu_{xx}+ u_{yy}= 0\), i.e. the solutions of the equation \({\mathfrak I}u= \delta(x- a,y)\), where \(\delta(x- a,y)\) is the Dirac measure concentrated at \((a,0)\). Since the Tricomi equation changes type in any neighborhood of \((a,0)\), the authors also analyze the influence that both the elliptic and hyperbolic parts have on the fundamental solutions. In view of the invariance of the Tricomi equation by translations along the x-axis, it suffices to determine fundamental solutions with pole at the origin. Let \(D_+\) be the region in the \((x,y)\) plane defined by \(D_+= \{(x,y)\in R^2:9x^2+ 4y^3> 0\}\) and let \(D_-\) be defined by \(D_-= \{(x,y)\in R^2:9x^2+ 4y^3< 0\}\). The following results are proved: Theorem 1.1 The distribution \(F_-= C_- E_-(x, y)\), where \[ E_+(x,y)= \begin{cases} (9x^2+ 4y^3)^{-1/6}\quad\text{in }D_+\\ 0\qquad \text{elsewhere}\end{cases} \] and \[ C_+= {\Gamma(1/6)\over 3.2^{2/3} \pi^{1/2}\Gamma(2/3)}, \] is a fundamental solution of the Tricomi operator. Theorem 1.2 The distribution \(F_+= C_+E_+(x,y)\), where \[ E_-= \begin{cases} |9x^2+ 4y^3|^{-1/6}\quad\text{in }D_-\\ 0\qquad \text{elsewhere}\end{cases} \] and \[ C_-= {3\Gamma(4/3)\over 2^{2/3} \pi^{1/2}\Gamma(5/6)}, \] is a fundamental solution of the Tricomi operator. The results of this paper can be extended, with small modifications, to the following generalizations of the Tricomi equation \(y^pu_{xx}+ u_{yy}= 0\) and \(y^p \sum^n_{i= 1} u_{x_ix_i}+ u_{yy}= 0\), where \(p\) is an odd integer.
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 35A08
zbMATH DE Number: 1425228
zbMATH Keywords: fundamental solution
zbMATH Keywords: Tricomi operator
zbMATH Keywords: partial differential equation of mixed type
zbMATH Keywords: generalizations of the Tricomi equation
author: J. Barros-Neto
author: I. M. Gel'fand
reviewed by: Elena Gavrilova
MaRDI profile type: Publication
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full work available at URL: s0012-7094-99-09814-9
OpenAlex ID: W1965039384
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description: scientific article; zbMATH DE number 1425228