# A differential inclusion: the case of an isotropic set

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- 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 (2010): 122-138. <http://eudml.org/doc/90753>.

@article{Croce2010,

abstract = {
In this article we are interested in the following problem: to
find a map $u: \Omega \to \mathbb\{R\}^2$ that satisfies
$$
\left\\{
\begin\{array\}\{ll\}
D u \in E\,\, &\mbox\{\{\it a.e.\} in \} \Omega\\ u(x)=\varphi(x) &x \in \partial \Omega \end\{array\}
\right.
$$
where Ω 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 φ.
},

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; rank one convex hull; isotropic set},

language = {eng},

month = {3},

number = {1},

pages = {122-138},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/90753},

volume = {11},

year = {2010},

}

TY - JOUR

AU - Croce, Gisella

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

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 $u: \Omega \to \mathbb{R}^2$ that satisfies
$$
\left\{
\begin{array}{ll}
D u \in E\,\, &\mbox{{\it a.e.} in } \Omega\\ u(x)=\varphi(x) &x \in \partial \Omega \end{array}
\right.
$$
where Ω 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 φ.

LA - eng

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

UR - http://eudml.org/doc/90753

ER -

## References

top- P. Cardaliaguet and R. Tahraoui, Equivalence between rank-one convexity and polyconvexity for isotropic sets of ${\mathbb{R}}^{2\times 2}$. I. Nonlinear Anal.50 (2002) 1179–1199.
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- B. Dacorogna and P. Marcellini, Implicit partial differential equations. Progr. Nonlinear Diff. Equ. Appl.37 (1999).
- B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case. Submitted.
- M. Gromov, Partial differential relations. Ergeb. Math. Grenzgeb.9 (1986).
- R.A. Horn and C.R. Johnson, Topics in matrix analysis. Cambridge University Press, Cambridge (1991).
- J. Kolář, Non-compact lamination convex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire20 (2003) 391–403.
- S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math.157 (2003) 715–742.
- R.T. Rockafellar, Convex analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1997). Reprint of the 1970 original, Princeton Paperbacks.

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