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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
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Full Text: Euclid