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

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.

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.

@article{AIF_1983__33_3_121_0,
author = {Spring, David},
title = {Convex integration of non-linear systems of partial differential equations},
journal = {Annales de l'Institut Fourier},
pages = {121--177},
publisher = {Institut Fourier},
volume = {33},
number = {3},
year = {1983},
doi = {10.5802/aif.934},
zbl = {0507.35019},
mrnumber = {85i:58126},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.934/}
}
TY  - JOUR
AU  - Spring, David
TI  - Convex integration of non-linear systems of partial differential equations
JO  - Annales de l'Institut Fourier
PY  - 1983
DA  - 1983///
SP  - 121
EP  - 177
VL  - 33
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.934/
UR  - https://zbmath.org/?q=an%3A0507.35019
UR  - https://www.ams.org/mathscinet-getitem?mr=85i:58126
UR  - https://doi.org/10.5802/aif.934
DO  - 10.5802/aif.934
LA  - en
ID  - AIF_1983__33_3_121_0
ER  - 
Spring, David. Convex integration of non-linear systems of partial differential equations. Annales de l'Institut Fourier, Tome 33 (1983) no. 3, pp. 121-177. doi : 10.5802/aif.934. http://archive.numdam.org/articles/10.5802/aif.934/

[1] M.L. Gromov, Convex Integration of Differential Relations, Math. USSR. Izvestia, 7 (1973), 329-343. | MR 54 #1323 | Zbl 0281.58004

[2] M.L. Gromov and J. Eliasberg, Removal of Singularities of Smooth Mappings, Math. USSR Izvestia, 5 (1971), 615-638. | MR 46 #394 | Zbl 0249.58005

[3] M.L. Gromov, Isometric Imbeddings and Immersions, Soviet Math. Dokl., 11 (1970), 794-797. | MR 43 #1212 | Zbl 0214.50404

[4] M.L. Gromov, Notes on Immersion Theory, I.H.E.S. (1981).

[5] M. Hirsch, Immersions of Manifolds, Trans. Amer. Math. Soc., 93 (1959), 242-276. | MR 22 #9980 | Zbl 0113.17202

[6] L. Khamam, Elimination géométrique des singularités avec applications aux équations aux dérivées partielles, thèse de 3e cycle, Université de Provence à Marseille, (1978).

[7] N.H. Kuiper, On C1 Isometric Imbeddings I, Nederl. Akad. Wet. Proc., Ser. A-58 (1955), 545-556. | MR 17,782c | Zbl 0067.39601

[8] J. Nash, On C1 Isometric Imbeddings, Annals of Math., 60 (1954), 383-396. | MR 16,515e | Zbl 0058.37703

[9] D. Spring, Convex Integration of Non-Linear Systems of Partial Differential Equations (preprint) (1979).

Cité par Sources :