Fortuitous relations between some elliptic equations

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Résumé: In this note we show that, for every solution $u = u(x,y)$ of some special two-dimensional elliptic equations in divergence form, some linear combinations of the derivatives $\displaystyle{\partial u \over \partial x}$ and $\displaystyle{\partial u \over \partial y}$ belong to $W^{1,2}_{loc}$ and solve other elliptic equations in divergence form (Theorems 1 and 2).
Moreover the associated equations for the derivatives enjoy the same ellipticity bounds and simultaneously converge with the initial one in the sense of $H$-convergence (Propositions 1 and 2).

Our results are mainly two-dimensional, but we give some partial extensions of them to dimension $n \geq 3$ in Section 4, where an application to a problem in non divergence form is also given.

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Date: 2003-12-01