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.

Keywords

  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

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal., 100 (1987), pp. 13--52.
2.
A. Capella and F. Otto, A rigidity result for a perturbation of the geometrically linear three-well problem, Comm. Pure Appl. Math., 62 (2009), pp. 1632--1669.
3.
A. Capella and F. Otto, A quantitative rigidity result for the cubic-to-tetragonal phase transition in the geometrically linear theory with interfacial energy, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), pp. 273--327, https://doi.org/10.1017/S0308210510000478.
4.
P. Cesana, M. Porta, and T. Lookman, Asymptotic analysis of hierarchical martensitic microstructure, J. Mech. Phys. Solids, 72 (2014), pp. 174--192.
5.
M. Chermisi and S. Conti, Multiwell rigidity in nonlinear elasticity, SIAM J. Math. Anal., 42 (2010), pp. 1986--2012.
6.
A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore, Harmonic analysis of the space BV, Rev. Mat. Iberoam., 19 (2003), pp. 235--263.
7.
S. Conti, Quasiconvex functions incorporating volumetric constraints are rank-one convex, J. Math. Pures Appl. (9), 90 (2008), pp. 15--30.
8.
S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Ration. Mech. Anal., 178 (2005), pp. 125--148.
9.
B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Vol. 37, Springer Science & Business Media, New York, 2012.
10.
B. Dacorogna, P. Marcellini, and E. Paolini, An explicit solution to a system of implicit differential equations, Ann. Inst. H. Poincare Non Linear Anal., 25 (2008), pp. 163--171.
11.
B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: Rigid maps and origami, J. Math. Pures Appl. (9), 90 (2008), pp. 66--81.
12.
B. Dacorogna, P. Marcellini, and E. Paolini, On the $n$-dimensional Dirichlet problem for isometric maps, J. Funct. Anal., 255 (2008), pp. 3274--3280.
13.
B. Dacorogna, P. Marcellini, and E. Paolini, Origami and partial differential equations, Notices Amer. Math. Soc., 57 (2010), pp. 598--606.
14.
G. Dolzmann and S. Müller, The influence of surface energy on stress-free microstructures in shape memory alloys, Meccanica, 30 (1995), pp. 527--539, https://doi.org/10.1007/BF01557083.
15.
G. Dolzmann and S. Müller, Microstructures with finite surface energy: The two-well problem, Arch. Ration. Mech. Anal., 132 (1995), pp. 101--141, https://doi.org/10.1007/BF00380505.
16.
G. H. Golub and C. F. Van Loan, Matrix Computations, Vol. 3, JHU Press, Baltimore, MD, 2012.
17.
M. L. Gromov, Convex integration of differential relations. i, Izv. Math., 7 (1973), pp. 329--343.
18.
R. L. Jerrard and A. Lorent, On multiwell Liouville theorems in higher dimension, Adv. Calc. Var., 6 (2013), pp. 247--298, https://doi.org/10.1515/acv-2012-0101.
19.
B. Kirchheim, Rigidity and Geometry of Microstructures, MPI-MIS lecture notes, 2003.
20.
B. Kirchheim, E. Spadaro, and L. Székelyhidi Jr., Equidimensional isometric maps, Comment. Math. Helv., 90 (2015), pp. 761--798.
21.
Y. Kitano and K. Kifune, HREM study of disclinations in MgCd ordered alloy, Ultramicroscopy, 39 (1991), pp. 279--286.
22.
S. Müller and V. Šverák, Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc., 1 (1999), pp. 393--422, http://dx.doi.org/10.1007/s100970050012.
23.
A. Rüland, The cubic-to-orthorhombic phase transition: Rigidity and non-rigidity properties in the linear theory of elasticity, Arch. Ration. Mech. Anal., 221 (2016), pp. 23--106, https://doi.org/10.1007/s00205-016-0971-5.
24.
A. Rüland, C. Zillinger, and B. Zwicknagl, Higher Sobolev Regularity of Convex Integration Solutions in Elasticity, preprint, https://arxiv.org/abs/1610.02529, 2016.
25.
T. Simon, Rigidity of Branching Microstructures in Shape Memory Alloys, preprint, https://arxiv.org/abs/1705.03664, 2017.
26.
L. Székelyhidi Jr., From Isometric Embeddings to Turbulence, MPI lecture notes, 2012.

Information & Authors

Information

Published In

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

History

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

Keywords

  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

Authors

Affiliations

Funding Information

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

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.