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

TL;DR

This paper develops a framework to establish higher Sobolev regularity for convex integration solutions in elasticity, applicable to both linear and nonlinear differential inclusions with affine boundary data.

Contribution

It introduces a general method to obtain higher Sobolev regularity for solutions to differential inclusions in elasticity, independent of boundary data position.

Findings

Higher integrability and differentiability exponents have a positive lower bound.

The framework applies to regularity of weak isometric immersions in 2D and 3D.

It addresses phase transformations in shape memory alloys.

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 $\text{int}(K^{lc})$. 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 $\text{int}(K^{lc})$. 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.