# A differential inclusion : the case of an isotropic set

ESAIM: Control, Optimisation and Calculus of Variations (2005)

- Volume: 11, Issue: 1, page 122-138
- ISSN: 1292-8119

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topCroce, Gisella. "A differential inclusion : the case of an isotropic set." ESAIM: Control, Optimisation and Calculus of Variations 11.1 (2005): 122-138. <http://eudml.org/doc/245592>.

@article{Croce2005,

abstract = {In this article we are interested in the following problem: to find a map $u: \Omega \rightarrow \mathbb \{R\}^2$ that satisfies\[ \left\lbrace \begin\{array\}\{ll\} D u \in E\,\, &\mbox\{\{\it a.e.\} in \} \Omega \vspace\{2.84544pt\}\\ u(x)=\varphi (x) &x \in \partial \Omega \hspace\{85.35826pt\} \end\{array\} \right. \]where $\Omega $ is an open set of $\mathbb \{R\}^2$ and $E$ is a compact isotropic set of $\mathbb \{R\}^\{2\times 2\}$. We will show an existence theorem under suitable hypotheses on $\varphi $.},

author = {Croce, Gisella},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {rank one convex hull; polyconvex hull; differential inclusion; isotropic set; differential inclusions; boundary value problem},

language = {eng},

number = {1},

pages = {122-138},

publisher = {EDP-Sciences},

title = {A differential inclusion : the case of an isotropic set},

url = {http://eudml.org/doc/245592},

volume = {11},

year = {2005},

}

TY - JOUR

AU - Croce, Gisella

TI - A differential inclusion : the case of an isotropic set

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2005

PB - EDP-Sciences

VL - 11

IS - 1

SP - 122

EP - 138

AB - In this article we are interested in the following problem: to find a map $u: \Omega \rightarrow \mathbb {R}^2$ that satisfies\[ \left\lbrace \begin{array}{ll} D u \in E\,\, &\mbox{{\it a.e.} in } \Omega \vspace{2.84544pt}\\ u(x)=\varphi (x) &x \in \partial \Omega \hspace{85.35826pt} \end{array} \right. \]where $\Omega $ is an open set of $\mathbb {R}^2$ and $E$ is a compact isotropic set of $\mathbb {R}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on $\varphi $.

LA - eng

KW - rank one convex hull; polyconvex hull; differential inclusion; isotropic set; differential inclusions; boundary value problem

UR - http://eudml.org/doc/245592

ER -

## References

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- [2] G. Croce, Ph.D. Thesis (2004).
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- [5] M. Gromov, Partial differential relations. Ergeb. Math. Grenzgeb. 9 (1986). Zbl0651.53001MR864505
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- [7] J. Kolář, Non-compact lamination convex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003) 391–403. Zbl1038.26008
- [8] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. 157 (2003) 715–742. Zbl1083.35032
- [9] R.T. Rockafellar, Convex analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1997). Reprint of the 1970 original, Princeton Paperbacks. Zbl0932.90001MR1451876

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