A differential inclusion : the case of an isotropic set

Gisella Croce

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

  • 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 (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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  1. [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. [2] G. Croce, Ph.D. Thesis (2004). 
  3. [3] B. Dacorogna and P. Marcellini, Implicit partial differential equations. Progr. Nonlinear Diff. Equ. Appl. 37 (1999). Zbl0938.35002MR1702252
  4. [4] B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case. Submitted. Zbl1107.49008
  5. [5] M. Gromov, Partial differential relations. Ergeb. Math. Grenzgeb. 9 (1986). Zbl0651.53001MR864505
  6. [6] R.A. Horn and C.R. Johnson, Topics in matrix analysis. Cambridge University Press, Cambridge (1991). Zbl0729.15001MR1091716
  7. [7] J. Kolář, Non-compact lamination convex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003) 391–403. Zbl1038.26008
  8. [8] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. 157 (2003) 715–742. Zbl1083.35032
  9. [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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