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.


  1. higher Sobolev regularity
  2. convex integration
  3. differential inclusion
  4. elasticity
  5. martensitic phase transitions
  6. weak isometric immersions

MSC codes

  1. 35B65
  2. 35B36
  3. 74B20
  4. 74N15

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 3791 - 3841
ISSN (online): 1095-7154


Submitted: 2 October 2017
Accepted: 20 February 2018
Published online: 12 July 2018


  1. higher Sobolev regularity
  2. convex integration
  3. differential inclusion
  4. elasticity
  5. martensitic phase transitions
  6. weak isometric immersions

MSC codes

  1. 35B65
  2. 35B36
  3. 74B20
  4. 74N15



Funding Information

Christ Church
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : CRC1060

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