Reducing real almost-linear second-order partial differential operators in two independent variables to a canonical form. (English) Zbl 0838.35030

Kozera, Ryszard

Czech. Math. J. 45, No. 3, 481-490 (1995).

This paper deals with the classical method of reducing real almost-linear second-order partial differential operators in two independent variables to a canonical form. In the present article, we intend to illuminate the geometrical character of those calculations. We first show that reducing a real almost-linear second-order partial differential operator to a canonical form amounts to reducing a suitable symmetric 2-contravariant tensor field to a canonical form. Next, we show that any symmetric 2-contravariant tensor field of locally constant type can locally be reduced to a canonical form. More specifically, given an almost-linear second-order partial differential operator \(P\) on an open region \(\Omega\) of \(\mathbb{R}^2\), \[ Pu= a_{11} {\partial^2 u\over \partial x^2_1}+ 2a_{12} {\partial^2 u\over \partial x_1\partial x_2}+ a_{22} {\partial^2 u\over \partial x^2_2}+ f\Biggl(x, u, {\partial u\over \partial x_1}, {\partial u\over \partial x_2}\Biggr)\;(u\in C^\infty_\mathbb{R}(\Omega), x\in \Omega), \] where \(a_{11}, a_{12}, a_{22}\in C^\infty_\mathbb{R}(\Omega)\) and \(f\in C^\infty_\mathbb{R}(\Omega\times \mathbb{R}^3)\), we associate with \(P\) a symmetric 2-contravariant tensor field \[ \sigma_{\mathring P}= a_{11} {\partial\over \partial x_1}\otimes_s {\partial\over \partial x_1}+ 2a_{12} {\partial\over \partial x_1}\otimes_s {\partial\over \partial x_2}+ a_{22} {\partial\over \partial x_2}\otimes_s {\partial\over \partial x_2}. \] We show that a canonical form of \(P\) can be found by reducing \(\sigma_{\mathring P}\) to a canonical form, and that the latter reduction can always be done locally in \(\Omega\) provided the type of \(\sigma_{\mathring P}\) is locally constant.

MSC:

35G20	Nonlinear higher-order PDEs
35A22	Transform methods (e.g., integral transforms) applied to PDEs

Keywords:

almost-linear second-order partial differential operators; canonical form

References:

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