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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.
##### MSC:
 35A25 Other special methods applied to PDEs 26B10 Implicit function theorems, Jacobians, transformations with several variables 35C99 Representations of solutions to partial differential equations
multiderivation