# The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 15, Issue: 2, page 322-366
- ISSN: 1292-8119

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topLorent, Andrew. "The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions." ESAIM: Control, Optimisation and Calculus of Variations 15.2 (2008): 322-366. <http://eudml.org/doc/90916>.

@article{Lorent2008,

abstract = {
Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\dots SO\left(2\right)A_\{N\}$
where $A_1,A_2,\dots, A_\{N\}$ are matrices of non-zero determinant. We
establish a sharp relation between the following two minimisation
problems in two dimensions. Firstly the N-well problem with surface energy. Let
$p\in\left[1,2\right]$, $\Omega\subset \mathbb\{R\}^2$ be a convex polytopal region. Define
$$
I^p\_\{\epsilon\}\left(u\right)=\int\_\{\Omega\} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2
u\left(z\right)\right|^2 \{\rm d\}L^2 z
$$
and let AF denote the subspace of functions in
$W^\{2,2\}\left(\Omega\right)$ that satisfy the affine boundary condition
Du=F on $\partial \Omega$ (in the sense of trace), where $F\not\in
K$. We consider the scaling (with respect to ϵ) of
$$
m^p\_\{\epsilon\}:=\inf\_\{u\in A\_F\} I^p\_\{\epsilon\}\left(u\right).
$$
Secondly the finite element approximation to the N-well problem
without surface energy. We will show there exists a space of functions $\mathcal\{D\}_F^\{h\}$ where
each function $v\in \mathcal\{D\}_F^\{h\}$ is piecewise affine on a regular
(non-degenerate) h-triangulation and satisfies the affine boundary
condition v=lF on $\partial \Omega$ (where lF is affine with
$Dl_F=F$) such that for
$$
\alpha\_p\left(h\right):=\inf\_\{v\in \mathcal\{D\}\_F^\{h\}\}
\int\_\{\Omega\}d^p\left(Dv\left(z\right),K\right) \{\rm d\}L^2 z
$$
there exists positive constants $\mathcal\{C\}_1<1<\mathcal\{C\}_2$ (depending on
$A_1,\dots, A_\{N\}$, p) for which the following holds true
$$
\mathcal\{C\}\_1\alpha\_p\left(\sqrt\{\epsilon\}\right)\leq m^p\_\{\epsilon\}\leq
\mathcal\{C\}\_2\alpha\_p\left(\sqrt\{\epsilon\}\right) \text\{ for all \}\epsilon>0.
$$},

author = {Lorent, Andrew},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Two wells; surface energy; non-convex variational problem; crystalline microstructure; stored energy density function; quasiconvex hull},

language = {eng},

month = {6},

number = {2},

pages = {322-366},

publisher = {EDP Sciences},

title = {The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions},

url = {http://eudml.org/doc/90916},

volume = {15},

year = {2008},

}

TY - JOUR

AU - Lorent, Andrew

TI - The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/6//

PB - EDP Sciences

VL - 15

IS - 2

SP - 322

EP - 366

AB -
Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\dots SO\left(2\right)A_{N}$
where $A_1,A_2,\dots, A_{N}$ are matrices of non-zero determinant. We
establish a sharp relation between the following two minimisation
problems in two dimensions. Firstly the N-well problem with surface energy. Let
$p\in\left[1,2\right]$, $\Omega\subset \mathbb{R}^2$ be a convex polytopal region. Define
$$
I^p_{\epsilon}\left(u\right)=\int_{\Omega} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2
u\left(z\right)\right|^2 {\rm d}L^2 z
$$
and let AF denote the subspace of functions in
$W^{2,2}\left(\Omega\right)$ that satisfy the affine boundary condition
Du=F on $\partial \Omega$ (in the sense of trace), where $F\not\in
K$. We consider the scaling (with respect to ϵ) of
$$
m^p_{\epsilon}:=\inf_{u\in A_F} I^p_{\epsilon}\left(u\right).
$$
Secondly the finite element approximation to the N-well problem
without surface energy. We will show there exists a space of functions $\mathcal{D}_F^{h}$ where
each function $v\in \mathcal{D}_F^{h}$ is piecewise affine on a regular
(non-degenerate) h-triangulation and satisfies the affine boundary
condition v=lF on $\partial \Omega$ (where lF is affine with
$Dl_F=F$) such that for
$$
\alpha_p\left(h\right):=\inf_{v\in \mathcal{D}_F^{h}}
\int_{\Omega}d^p\left(Dv\left(z\right),K\right) {\rm d}L^2 z
$$
there exists positive constants $\mathcal{C}_1<1<\mathcal{C}_2$ (depending on
$A_1,\dots, A_{N}$, p) for which the following holds true
$$
\mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\epsilon}\leq
\mathcal{C}_2\alpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0.
$$

LA - eng

KW - Two wells; surface energy; non-convex variational problem; crystalline microstructure; stored energy density function; quasiconvex hull

UR - http://eudml.org/doc/90916

ER -

## References

top- L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal.139 (1997) 201–238. Zbl0890.49019
- L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). Zbl0957.49001
- J.M. Ball and R.D. James, Fine phase mixtures as minimisers of energy. Arch. Rational Mech. Anal.100 (1987) 13–52. Zbl0629.49020
- J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructure and the two well problem. Phil. Trans. Roy. Soc. London Ser. A338 (1992) 389–450. Zbl0758.73009
- M. Chipot, The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math.83 (1999) 325–352. Zbl0937.65070
- M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal.103 (1988) 237–277. Zbl0673.73012
- M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of non-convex problems, in Variations of domain and free-boundary problems in solid mechanics (Paris, 1997), Solid Mech. Appl.66, Kluwer Acad. Publ., Dordrecht (1999) 317–325.
- S. Conti, Branched microstructures: scaling and asymptotic self-similarity. Comm. Pure Appl. Math.53 (2000) 1448–1474. Zbl1032.74044
- S. Conti and B. Schweizer, Rigidity and Gamma convergence for solid-solid phase transitions with $SO\left(2\right)$-invariance. Comm. Pure Appl. Math.59 (2006) 830–868. Zbl1146.74018
- S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to ${L}^{1}$ estimates: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions. Arch. Rational Mech. Anal.175 (2005) 287–300. Zbl1080.49026
- S. Conti, G. Dolzmann and B. Kirchheim, Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions. Ann. Inst. H. Poincaré Anal. Non Linéaire24 (2007) 953–962. Zbl1131.74037
- B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Math.178 (1997) 1–37. Zbl0901.49027
- C. De Lellis and L. Székelyhidi, The Euler equations as a differential inclusion. Ann. Math. (to appear). Zbl05710190
- G. Dolzmann and K. Bhattacharya, Relaxed constitutive relations for phase transforming materials. The J. R. Willis 60th anniversary volume. J. Mech. Phys. Solids48 (2000) 1493–1517. Zbl0966.74053
- G. Dolzmann and B. Kirchheim, Liquid-like behavior of shape memory alloys. C. R. Math. Acad. Sci. Paris336 (2003) 441–446. Zbl1113.74411
- G. Dolzmann and S. Müller, Microstructures with finite surface energy: the two-well problem. Arch. Rational Mech. Anal.132 (1995) 101–141. Zbl0846.73054
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). Zbl0804.28001
- G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math.55 (2002) 1461–1506. Zbl1021.74024
- B. Kirchheim, Deformations with finitely many gradients and stability of quasiconvex hulls. C. R. Acad. Sci. Paris Sér. I Math.332 (2001) 289–294. Zbl0989.49013
- B. Kirchheim, Rigidity and Geometry of Microstructures. Lectures note 16/2003, Max Planck Institute for Mathematics in the Sciences, Leipzig (2003).
- R.V. Kohn, New Estimates for Deformations in Terms of Their Strains. Ph.D. thesis, Princeton University, USA (1979).
- R.V. Kohn and S. Müller, Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math.47 (1994) 405–435. Zbl0803.49007
- A. Lorent, An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure. ESAIM: M2AN35 (2001) 921–934. Zbl1017.74067
- M. Luskin, On the computation of crystalline microstructure. Acta Numer.5 (1996) 191–257. Zbl0867.65033
- P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics44. Cambridge University Press (1995). Zbl0819.28004
- S. Müller, Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differ. Equ.1 (1993) 169–204. Zbl0821.49015
- S. Müller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems (Cetraro, 1996), Lecture Notes in Mathematics1713, Springer, Berlin (1999) 85–210. www.mis.mpg.de/cgi-bin/lecturenotes.pl.
- S. Müller, Uniform Lipschitz estimates for extremals of singularly perturbed nonconvex functionals. MIS MPG, Preprint 2 (1999).
- S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, in Geometric Analysis and the Calculus of Variations. For Stefan Hildebrandt, J. Jost Ed., International Press, Cambridge (1996) 239–251. Zbl0930.35038
- S. Müller and V. Šverák, Convex integration with constraints and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. (JEMS)1 (1999) 393–422. Zbl0953.35042
- S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math.157 (2003) 715–742. Zbl1083.35032
- S. Müller, M. Rieger and V. Šverák, Parabolic systems with nowhere smooth solutions. Arch. Rational Mech. Anal.177 (2005) 1–20. Zbl1116.35059
- M.A. Sychev, Comparing two methods of resolving homogeneous differential inclusions. Calc. Var. Partial Differ. Equ.13 (2001) 213–229. Zbl0994.35038
- M.A. Sychev and S. Müller, Optimal existence theorems for nonhomogeneous differential inclusions. J. Funct. Anal.181 (2001) 447–475. Zbl0989.49012
- V. Šverák, On the problem of two wells, in Microstructure and phase transition, D. Kinderlehrer, R.D. James, M. Luskin and J. Ericksen Eds., IMA Vol. Math. Appl.54, Springer, New York (1993) 183–189. Zbl0797.73079

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