# A differential inclusion : the case of an isotropic set

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

top

top
In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^{2}$ that satisfies$\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \phantom{\rule{85.35826pt}{0ex}}\hfill \end{array}\right$/extract_itex]where $\Omega$ is an open set of ${ℝ}^{2}$ and $E$ is a compact isotropic set of ${ℝ}^{2×2}$. We will show an existence theorem under suitable hypotheses on $\varphi$. ## How to cite top 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

top
1. [1] P. Cardaliaguet and R. Tahraoui, Equivalence between rank-one convexity and polyconvexity for isotropic sets of ${ℝ}^{2×2}$. 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

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.