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Analysis and computations in the four-well problem. (English) Zbl 0866.49031
Kenmochi, N. (ed.) et al., Proceedings of the Banach Center minisemester on nonlinear analysis and applications, Warsaw, Poland, November 14 – December 15, 1994. Tokyo: Gakkōtosho Co., Ltd. GAKUTO Int. Ser., Math. Sci. Appl. 7, 67-78 (1995).
The authors consider vector problems of the calculus of variations of the form \[ \inf\Biggl\{\int_\Omega F(\nabla v)dxdy\mid v\in W^{1,\infty}_0(\Omega;R^2)\Biggr\}, \] where \(\Omega\) is a bounded domain in \(R^2\). The function \(F\) is a nonconvex continuous function from \(M_2\) (the space of \(2\times 2\) matrices) into \(R\), it admits four strict minima \(W_1\), \(W_2\), \(W_3\), \(W_4\) in \(M_2\) and \(F(W_1)=F(W_2)=F(W_3)=F(W_4)=0\). The four wells \(W_1\), \(W_2\), \(W_3\) and \(W_4\) are two by two rank-one incompatible. The above problem has no solution, its infimum is equal to zero, minimizing sequences \((v_h)_h\) satisfying \(|v_h|+ |\nabla v_h|\leq C\), uniformly converge to zero and define the unique Young measure \(\nu_x={1\over 4}\sum_{i=1}^4\delta_{W_i}\). Numerical experiments based on gradient methods are presented. It is highlighted that the method is very sensitive to the initial choice \(v_0\) and the descent direction is difficult to calculate when \(v\) is close to zero. Some directions are suggested to overcome these numerical difficulties.
For the entire collection see [Zbl 0853.00039].

90C52 Methods of reduced gradient type
65K10 Numerical optimization and variational techniques
49J40 Variational inequalities