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The two well problem for piecewise affine maps. (English) Zbl 1258.35061
Summary: In the two-well problem we look for a map $$u$$ which satisfies Dirichlet boundary conditions and whose gradient $$Du$$ assumes values in $$SO\left( 2\right) A\cup SO\left( 2\right) B=\mathbb{S}_{A}\cup \mathbb{S}_{B},$$ for two given invertible matrices $$A,B$$ (an element of $$SO\left( 2\right) A$$ is of the form $$RA$$ where $$R$$ is a rotation). In the original approach by J. M. Ball and R. D. James [Arch. Ration. Mech. Anal. 100, No. 1, 13–52 (1987; Zbl 0629.49020)], $$A, B$$ are two matrices such that $$\det B>\det A>0$$ and $$\mathrm{rank}\left\{ A-B\right\} =1.$$ It was proved in the 1990’s that a map $$u$$ satisfying given boundary conditions and such that $$Du\in\mathbb{S} _{A}\cup\mathbb{S}_{B}$$ exists in the Sobolev class $$W^{1,\infty} (\Omega;\mathbb{R}^{2})$$ of Lipschitz continuous maps. However, for orthogonal matrices it was also proved that solutions exist in the class of piecewise-$$C^{1}$$ maps, in particular in the class of piecewise-affine maps. We prove here that this possibility does not exist for other nonsingular matrices $$A, B$$: precisely, the two-well problem can be solved by means of piecewise-affine maps only for orthogonal matrices.

MSC:
 35F60 Boundary value problems for systems of nonlinear first-order PDEs 35F50 Systems of nonlinear first-order PDEs
Keywords:
Dirichlet boundary conditions
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