Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in int(K^lc)

2017/09/08 by Angkana Rüland, Rüland, Angkana, Christian Zillinger +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Contact Mechanics and Variational Inequalities #Elasticity and Material Modeling

paper · pdf · doi:10.48550/arxiv.1709.02880

Abstract

In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a general framework yielding higher Sobolev regularity for Dirichlet problems with affine data in int(K^lc). This allows us to simultaneously deal with linear and nonlinear differential inclusion problems. We show that the derived higher integrability and differentiability exponent has a lower bound, which is independent of the position of the Dirichlet boundary data in int(K^lc). As applications we discuss the regularity of weak isometric immersions in two and three dimensions as well as the differential inclusion problem for the geometrically linear hexagonal-to-rhombic and the cubic-to-orthorhombic phase transformations occurring in shape memory alloys.

Citations

• Fine phase mixtures as minimizers of energy

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