Geodesic
Unexpected solutions of first and second order partial differential equations
Documenta mathematica, ICM Berlin 1998, Vol. II (1998), pp. 691-702.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

This note discusses a general approach to construct Lipschitz solutions of $Du \in K$, where $u: \Omega \subset \bbfR^n \to \bbfR^m$ and where $K$ is a given set of $m\times n$ matrices. The approach is an extension of Gromov's method of convex integration. One application concerns variational problems that arise in models of microstructure in solid-solid phase transitions. Another application is the systematic construction of singular solutions of elliptic systems. In particular, there exists a $2 \times 2$ (variational) second order strongly elliptic system $ {div}\sigma(Du) = 0 $ that admits a Lipschitz solution which is nowhere $C^1$.
Classification : 35F30, 35J55, 74B20, 49J24
Keywords: phase transitions, regularity, singular solutions, microstructure, convex integration 
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     author = {M\"uller, Stefan and \v{S}ver\'ak, Vladimir},
     title = {Unexpected solutions of first and second order partial differential equations},
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     volume = {ICM Berlin 1998, Vol. II},
     year = {1998},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_1998__S8__a16/}
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Müller, Stefan; Šverák, Vladimir. Unexpected solutions of first and second order partial differential equations. Documenta mathematica, ICM Berlin 1998, Vol. II (1998), pp. 691-702. http://geodesic.mathdoc.fr/item/DOCMA_1998__S8__a16/