A differential inclusion: the case of an isotropic set

Gisella Croce

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

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

Abstract

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In this article we are interested in the following problem: to find a map u : Ω 2 that satisfies D u E a.e. in Ω u ( x ) = ϕ ( x ) x Ω where Ω is an open set of 2 and E is a compact isotropic set of 2 × 2 . We will show an existence theorem under suitable hypotheses on φ.

How to cite

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Croce, 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

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  1. P. Cardaliaguet and R. Tahraoui, Equivalence between rank-one convexity and polyconvexity for isotropic sets of 2 × 2 . I. Nonlinear Anal.50 (2002) 1179–1199.  Zbl1004.49007
  2. G. Croce, Ph.D. Thesis (2004).  
  3. B. Dacorogna and P. Marcellini, Implicit partial differential equations. Progr. Nonlinear Diff. Equ. Appl.37 (1999).  Zbl0938.35002
  4. B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case. Submitted.  Zbl1107.49008
  5. M. Gromov, Partial differential relations. Ergeb. Math. Grenzgeb.9 (1986).  Zbl0651.53001
  6. R.A. Horn and C.R. Johnson, Topics in matrix analysis. Cambridge University Press, Cambridge (1991).  Zbl0729.15001
  7. J. Kolář, Non-compact lamination convex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire20 (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.  

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