Convex integration of non-linear systems of partial differential equations

David Spring

Annales de l'institut Fourier (1983)

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

Abstract

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Geometrical techniques are employed to prove a global existence theorem for C r -solutions to underdetermined systems of non-linear r t h order partial differential equations, r ∈ { 1 , 2 , 3 , ... } , 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.

How to cite

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Spring, 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

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  1. [1] M.L. GROMOV, Convex Integration of Differential Relations, Math. USSR. Izvestia, 7 (1973), 329-343. Zbl0281.58004MR54 #1323
  2. [2] M.L. GROMOV and J. ELIASBERG, Removal of Singularities of Smooth Mappings, Math. USSR Izvestia, 5 (1971), 615-638. Zbl0249.58005MR46 #394
  3. [3] M.L. GROMOV, Isometric Imbeddings and Immersions, Soviet Math. Dokl., 11 (1970), 794-797. Zbl0214.50404MR43 #1212
  4. [4] M.L. GROMOV, Notes on Immersion Theory, I.H.E.S. (1981). 
  5. [5] M. HIRSCH, Immersions of Manifolds, Trans. Amer. Math. Soc., 93 (1959), 242-276. Zbl0113.17202MR22 #9980
  6. [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. [7] N.H. KUIPER, On C1 Isometric Imbeddings I, Nederl. Akad. Wet. Proc., Ser. A-58 (1955), 545-556. Zbl0067.39601MR17,782c
  8. [8] J. NASH, On C1 Isometric Imbeddings, Annals of Math., 60 (1954), 383-396. Zbl0058.37703MR16,515e
  9. [9] D. SPRING, Convex Integration of Non-Linear Systems of Partial Differential Equations (preprint) (1979). 

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