# Convex integration of non-linear systems of partial differential equations

Annales de l'institut Fourier (1983)

- Volume: 33, Issue: 3, page 121-177
- ISSN: 0373-0956

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topSpring, David. "Convex integration of non-linear systems of partial differential equations." Annales de l'institut Fourier 33.3 (1983): 121-177. <http://eudml.org/doc/74593>.

@article{Spring1983,

abstract = {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\!\in \!\lbrace 1,2,3,\!\ldots \rbrace $, 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.},

author = {Spring, David},

journal = {Annales de l'institut Fourier},

keywords = {convex integration; global existence; underdetermined systems; convexity conditions},

language = {eng},

number = {3},

pages = {121-177},

publisher = {Association des Annales de l'Institut Fourier},

title = {Convex integration of non-linear systems of partial differential equations},

url = {http://eudml.org/doc/74593},

volume = {33},

year = {1983},

}

TY - JOUR

AU - Spring, David

TI - Convex integration of non-linear systems of partial differential equations

JO - Annales de l'institut Fourier

PY - 1983

PB - Association des Annales de l'Institut Fourier

VL - 33

IS - 3

SP - 121

EP - 177

AB - 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\!\in \!\lbrace 1,2,3,\!\ldots \rbrace $, 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.

LA - eng

KW - convex integration; global existence; underdetermined systems; convexity conditions

UR - http://eudml.org/doc/74593

ER -

## References

top- [1] M.L. GROMOV, Convex Integration of Differential Relations, Math. USSR. Izvestia, 7 (1973), 329-343. Zbl0281.58004MR54 #1323
- [2] M.L. GROMOV and J. ELIASBERG, Removal of Singularities of Smooth Mappings, Math. USSR Izvestia, 5 (1971), 615-638. Zbl0249.58005MR46 #394
- [3] M.L. GROMOV, Isometric Imbeddings and Immersions, Soviet Math. Dokl., 11 (1970), 794-797. Zbl0214.50404MR43 #1212
- [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. Zbl0113.17202MR22 #9980
- [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. Zbl0067.39601MR17,782c
- [8] J. NASH, On C1 Isometric Imbeddings, Annals of Math., 60 (1954), 383-396. Zbl0058.37703MR16,515e
- [9] D. SPRING, Convex Integration of Non-Linear Systems of Partial Differential Equations (preprint) (1979).

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