Convex integration of non-linear systems of partial differential equations
Annales de l'Institut Fourier, Volume 33 (1983) no. 3, pp. 121-177.

Geometrical techniques are employed to prove a global existence theorem for ${C}^{r}$-solutions to underdetermined systems of non-linear ${r}^{th}$ order partial differential equations, $r\phantom{\rule{-0.166667em}{0ex}}\in \phantom{\rule{-0.166667em}{0ex}}\left\{1,2,3,\phantom{\rule{-0.166667em}{0ex}}...\right\}$, which satisfy certain convexity conditions. The solutions are not unique, but satisfy given approximations on lower order derivatives. The main result, which includes the relative case generalizes the work of M. Gromov on non-linear first order systems.

Des techniques topologiques sont employées pour démontrer un théorème d’existence globale de ${C}^{r}$-solutions aux systèmes non-linéaires des équations aux dérivées partielles d’ordre $r,r\phantom{\rule{-0.166667em}{0ex}}\in \phantom{\rule{-0.166667em}{0ex}}\left\{1,2,3,\phantom{\rule{-0.166667em}{0ex}}...\right\}.$ Ces systèmes sont sous-déterminés et doivent satisfaire certaines conditions de convexité. Les solutions ne sont pas uniques mais elles satisfont certaines approximations sur les dérivées d’ordre inférieur. Le résultat principal, qui comporte aussi le cas relatif, généralise les travaux de M. Gromov sur les systèmes non-linéaires d’ordre 1.

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Spring, David. Convex integration of non-linear systems of partial differential equations. Annales de l'Institut Fourier, Volume 33 (1983) no. 3, pp. 121-177. doi : 10.5802/aif.934. http://archive.numdam.org/articles/10.5802/aif.934/

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