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On a problem of potential wells. (English) Zbl 0880.49005

This paper gives a result for the existence of a minimizer for an energy functional of the kind \(\int_\Omega g(\nabla u(x))dx\), where \(g\) is nonnegative and is zero only on potential wells described by rotations of finitely many matrices \(A_1,\dots,A_r\). The problem of finding a minimizer is than equivalent to solving the differential inclusion \[ \nabla u(x)\in\bigcup^r_{i=1} \text{SO}(3)A_i, \] where \(\text{SO}(3)\) describe rotations of matrices \(A_i\). In the paper, it is proved that for any open and bounded set \(\Omega\subset\mathbb{R}^3\) the problem \[ \nabla u(x)\in \text{SO}(3)I\cup \text{SO}(3)I^-,\quad|u|_{\partial\Omega}= 0, \] where \[ I^-=\begin{pmatrix} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix} \] has a solution.

MSC:

49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
34A60 Ordinary differential inclusions
49J10 Existence theories for free problems in two or more independent variables
74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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