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On the multiderivation of Antonio Mambriani. (Italian) Zbl 0798.35007
The author begins by recalling the following definition [A. Mambriani, Riv. Mat. Univ. Parma 6, 321-348 (1955; Zbl 0067.319)]: A pluridifferentiator is an operator \({\mathcal D}= \sum^ n_{i=1} X_ i(x)\partial/\partial x_ i\), where \(X_ i(x)\) \((i= 1,\dots,n)\) are functions continuous in a domain \(B\subseteq {\mathbb{R}}^ n\) along with their first partial derivatives. She then applies pluridifferentiators to particular types of first- and second-order linear partial differential equations. For the first order she studies the so-called separated- variable equation \[ {\mathcal D} z(x)= f(x)/g(z(x); \omega_ 2(x),\dots,\omega_ n(x)),\tag{*} \] where \(z(x)\) is the unknown function and \(f(x)\), \(g(z;\omega_ 2,\dots,\omega_ n)\) are given continuous functions with continuous partial derivatives with respect to \(\omega_ 2,\dots,\omega_ n\), as well as linear equations of the form \(({\mathcal D}+ A(x))z(x)= f(x)\), \(x\in B\), where \(A(x)\), \(f(x)\) are given functions continuous in \(B\). The author shows that, if \(G_ *(z(x); \omega_ 2(x),\dots,\omega_ n(x))= \int_ * g(z(x); \omega_ 2(x),\dots,\omega_ n(x))dz\) is a certain “determination” of the indefinite integral, then the general integral of \((*)\) is represented by \(G_ *(z(x); \omega_ 2(x),\dots,\omega_ n(x))\).
For the application to some known second-order equations the author considers the pluridifferentiator \[ {\mathcal L}= a_{11} \partial^ 2/\partial x^ 2+ 2a_{12} \partial^ 2/\partial x\partial y+ a_{22} \partial^ 2/\partial y^ 2+ 2a_{13} \partial/\partial x+ 2a_{23} \partial/\partial y+ a_{33}, \] with constant coefficients \(a_{ij}= a_{ji}\), and shows that it is decomposable into the product of two distinct first-order nonhomogeneous linear pluridifferentiators with constant coefficients if and only if the determinant \(A= | a_{ij}|\) is zero but not all of its second-order minors are zero. Therefore, in the hyperbolic case: \({\mathcal L}= \partial^ 2/\partial x^ 2- a^ 2 \partial^ 2/\partial y^ 2\), and in the elliptic case: \({\mathcal L}= \partial^ 2/\partial x^ 2+ a^ 2 \partial^ 2/\partial y^ 2\), the pluridifferentiator is decomposable into the product of two distinct first-order pluridifferentiators, whereas in the parabolic case: \({\mathcal L}= \partial^ 2/\partial x^ 2- b\partial/\partial y\), this is not possible.
35A25 Other special methods applied to PDEs
26B10 Implicit function theorems, Jacobians, transformations with several variables
35C99 Representations of solutions to partial differential equations