In this paper, we discuss higher Sobolev regularity of convex integration solutions for the geometrically nonlinear two-well problem. More precisely, we construct solutions to the differential inclusion [Formula: see text] subject to suitable affine boundary conditions for [Formula: see text] with [Formula: see text] such that the associated deformation gradients [Formula: see text] enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where [Formula: see text], and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the nonlinear matrix space geometry, it is possible to deal with the geometrically nonlinear two-well problem within the framework outlined in [A. Rüland, C. Zillinger and B. Zwicknagl, Higher Sobolev regularity of convex integration solutions in elasticity: The Dirichlet problem with affine data in int[Formula: see text], SIAM J. Math. Anal. 50 (2018) 3791–3841]. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures.

中文翻译：

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具有较高 Sobolev 正则性的几何非线性二井问题的凸积分解

在本文中，我们讨论了几何非线性两井问题凸积分解的更高 Sobolev 正则性。更准确地说，我们构造微分包含 [公式：见文本] 的解，并满足 [公式：见文本] 和 [公式：见文本] 的合适仿射边界条件，使得相关的变形梯度 [公式：见文本] 享有更高索博列夫规律。这为形状记忆合金中的相变建模提供了第一个结果，其中 [公式：参见文本]，并且通过凸积分构造的能量最小化器满足更高的 Sobolev 规则。我们表明，尽管非线性矩阵空间几何的处理带来了额外的困难，在 [A. Rüland、C. Zillinger 和 B. Zwicknagl，弹性中凸积分解决方案的更高 Sobolev 规律性：具有 int 仿射数据的 Dirichlet 问题[公式：见正文]，SIAM J. Math。肛门。50 (2018) 3791–3841]。在物理上，我们对更高 Sobolev 规则的凸积分解决方案的研究是通过将规则视为微观结构的可能选择机制来推动的。