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

Cellina, Arrigo; Perrotta, Stefania

J. Convex Anal. 2, No. 1-2, 103-115 (1995).

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.

Reviewer: V.Křivan (České Budějovice)

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

Keywords:

minimizer for an energy functional; potential wells; differential inclusion; rotations of matrices