ISSN:
1432-0835
Keywords:
Mathematics Subject Classification (1991):49J45, 49Q20, 49N60, 73T05, 73V30
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. The integral representation for the relaxation of a class of energy functionals where the admissible fields are constrained to remain on a $C^1$ $m$ -dimensional manifold $\mathcal{M}\subset \mathbb{ R}^d $ is obtained. If $f: \mathbb{ R}^{d \times N} \to [0, \infty)$ is a continuous function satisfying $0 \le f(\xi) \le C (1 +{|\xi|}^p),$ for $ C 〉 0, \,p \geq 1,$ and for all $\xi \in \mathbb{ R}^{d \times N},$ then \begin{eqnarray} \mathcal{F}(u, \Omega) :&=& \inf_{\{u_n\}} \left\{\! \mathop{\rm inf}\limits_{\{ u_n\}} \int_\Omega f(\nabla u_n) \, dx: u_n \rightharpoonup u \, \mbox{in}\, \, W^{1,p}, \right. \nonumber && \left. u_n (x) \in \mathcal{M}\,\, \mbox{a.e.}\,\, x \in \Omega, n \in{\mathbb{N}} \! \right\}\nonumber &=& \int_\Omega Q_T f(u, \nabla u) \, dx, \nonumber \end{eqnarray} where $\Omega \subset \mathbb{ R}^N $ is open, bounded, and $Q_T f(y_0, \xi)$ is the tangential quasiconvexification of f at $y_0 \in \mathcal{M},$ $\xi = ({\xi}^1,...,{\xi}^N),{\xi}^i $ belong to the tangent space to $\mathcal{M}$ at $y_0,\, i=1,...,N.$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s005260050137
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