Fundamental solutions for the Tricomi operator
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.
partial differential equation of mixed type
scientific article; zbMATH DE number 1425228
scientific article