# zbMATH — the first resource for mathematics

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].

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