Gellerstedt and Laplace-Beltrami operators relative to a mixed signature metric
In the authors' papers [(*) Adv. Differ. Equ. 10, No. 4, 445--461 (2005; Zbl 1102.35073) and Selecta Math. New Ser. 11, 309--324 (2005; Zbl 1111.35011)] fundamental solutions to generalized Tricomi operators in \(\mathbb{R}^{n+1}\) \((\geq 1)\) were discussed and there the operators \[ Q_\alpha:={1\over 2} \Delta_{x,s}+ {\alpha\over s} {\partial\over\partial s},\quad R_\alpha:={1\over 2}\square_{x,t}+ {\alpha\over t} {\partial\over\partial t},\quad\alpha\in C\mid\{0\}, \] with \(x= (x_1,\dots, x_n)\), \(s= x_{n+1}\), and \(t= x_{n+1}\) arose in a natural way. They are first-order singular perturbations of the Laplace \(\Delta_{x,s}= \partial^2/\partial s^2- \sum^n_{i=1}\partial^2/\partial x^2_i\) and D'Alembertian \(\square_{x,t}= \partial^2/\partial t^2-\sum^n_{i=1} \partial^2/\partial x^2_i\) operators. Actually the hyperbolic operator \(R_\alpha\) is the Euler-Poisson-Darboux operator studied by \textit{S. Delache} and \textit{J. Leray} in [Bull. Soc. Math. Fr. 99, 313--336 (1971; Zbl 0209.12701)], where they described its fundamental solutions. Their work constitutes the basis of the investigations in [(*)]. The elliptic operator \(Q_\alpha\) was studied by \textit{J. B. Diaz} and \textit{A. Weinstein} in [On the fundamental solution of a singular Beltrami operator. Studies Math. Mech., presented to Richard von Mises, 97--102 (1954; Zbl 0057.08003)] and by \textit{A. Weinstein} in [Commun. Pure Appl. Math. 7, 105--116 (1954; Zbl 0056.09301)]. In the present paper Gellerstedt and Laplace-Beltrami operators are under consideration. They have an interesting and important property: under suitable change of coordinates they can be represented, up to a multiplying factor, in terms of \(Q_\alpha\), and \(R_\alpha\) singular first-order perturbations of the Laplace and D'Alembertian operators. Knowledge of fundamental solutions for \(Q_\alpha\) and \(R_\alpha\) gives the possibility to find explicit formulas fundamental to those operators.
Gellerstedt and Laplace-Beltrami operators
scientific article; zbMATH DE number 5610637