Finally,γ→0finishes the proof of the lower bound.
condition has the same scaling as Rr →1.
As the constant for Korn’s inequality is invariant under dilations of the domain, we will only consider the caseR= 1, r <1.
The technique to prove an upper bound for the constant is a covering argument which is classical for proving Korn’s inequality for thin domains. We show first a uniform bound on the constant for special subsets of the annulus.
Lemma 5.A.1. For r <1 define the set Ur={t(sinθ,−cosθ) :r <1<1,|θ|< 1−r2 }. There exist r0<1 and a constantC >0 such that for every r0< r <1 and every u∈H1(Ur;R2)there exists a skew-symmetric matrixW such that
ˆ
Ur
|∇u−W|2dx≤C ˆ
Ur
|(∇u)sym|2dx.
Proof. Let us define the square U˜r = r−12 ,1−r2
×(0,1−r) and the function fr : ˜Ur → Ur by fr(θ, t) = (1−t)(sinθ,−cosθ), see figure 5.6. Clearly,fris a diffeomorphism and its derivative is
∇fr(t, θ) = (1−t) cosθ (1−t) sinθ
−sinθ cosθ
! .
It holds|∇fr(t, θ)−Id| ≤ O(r)where O(r)→0as r→1.
Now, letu∈H1(Ur;R2)such that ´
Ur(∇u)skewdx= 0. Defineu˜: ˜Ur→R2 by ˜u(θ, t) =u(fr(θ, t)).
As∇fris almost the identity matrix, the symmetric part of∇u˜is approximately(∇u)sym◦fr. Indeed, by the chain rule we find
|(∇˜u)sym−(∇u)sym◦fr| ≤ O(r)|((∇u)◦fr)sym| ≤ O(r)|(∇u)◦fr|. (5.36) By Korn’s inequality applied tou, there exists a skew-symmetric matrix˜ W and a constantK ≥1
such that ˆ
U˜r
|∇˜u−W|2dx≤K ˆ
U˜r
|(∇u)˜ sym|2dx.
Note that by scaling the constant does not depend onr. By (5.36), we find further ˆ
U˜r
|(∇u)˜ sym|2dx≤2 ˆ
U˜r
|(∇u)sym◦fr|2+O(r)2|(∇u)◦fr|2dx
= 2 ˆ
Ur
(|(∇u)sym|2+O(r)2|∇u|2)|det∇fr−1|.
On the other hand, one proves similarly that ˆ
U˜r
|∇˜u−W|2dx≥ 1 2 ˆ
Ur
(|∇u|2− O(r)2|∇u|2)|det∇fr−1|dx, where one uses that from´
Ur(∇u)skewdx= 0it follows´
Ur|∇u−W|2dx≥´
Ur|∇u|2dx.
Notice that for r close to 1 the gradient ∇fr is uniformly close to the identity. The same holds for∇fr−1. Chooser0 so close to 1 such that for all smaller r it holds that |det∇fr−1−1| ≤ 12 and
1−r fr
U˜r 1−r Ur
Figure 5.6: Sketch of the situation in Lemma 5.A.1.
O(r)≤ 1
4√
K. Then, one obtains, by combining the previous inequalities, ˆ
Ur
|∇u|2dx≤12K ˆ
Ur
|(∇u)sym|2dx+13 16
ˆ
Ω
|∇u|2dx.
Absorbing the very right term to the left hand side ends the proof.
Remark 5.A.1. The same holds true for any rotated version of Ur with the same constant since Korn’s constant does not depend on the choice of coordinates.
Remark 5.A.2. For an explicit bound on the constructed constant in the previous lemma, note that an upper bound for the optimal constant for the square was computed in [49], namely8 + 4√
2. In [48], it is conjectured that the optimal constant for a square is7.
In the following lemma, we state an upper bound on Korn’s constant for the annulusB1(0)\Br(0) forcurl-free fields.
Lemma 5.A.2. There existsr0<1such that for allr0< r <1Korn’s constantK(r)forB1(0)\Br(0) is less or equal than C(1−r)−2 where C >0 is a universal constant. Precisely, for every function β ∈ L2(B1(0)\Br(0);R2×2) satisfying curlβ = 0 there exists a skew-symmetric matrix W ∈ R2×2
such that ˆ
B1(0)\Br(0)
|β−W|2dx≤C(1−r)−2 ˆ
B1(0)\Br(0)
|βsym|2dx.
Proof. Letr0as in Lemma 5.A.1 and0< r < r0. In analogy to the previous lemma, we define the sets Urk =
t(cosθ,sinθ) :r < t <1 and k−1
2 (1−r)< θ < k+ 1 2 (1−r)
wherek= 1, . . . , 4π
1−r
=Lr.
As b
4π 1−rc+1
2 (1 − r) > 2π, it follows B1(0)\ Br(0) ⊂ SLr
k=1Urk. Moreover, notice that for all k= 1, . . . , Lr−1 it holds |Urk|
|Urk∩Urk+1| = 2andPLr
k=11Ukr ≤2.
Now, let β ∈ L2(B1(0)\Br(0);R2×2) such that curlβ = 0. In particular, β can be written as a gradient on each Ukr. For each k = 1, . . . , Lr, we apply Lemma 5.A.1 and Remark 5.A.1 on Urk to obtain a skew-symmetric matrixWk such that
ˆ
Urk
|β−Wk|2dx≤C ˆ
Urk
|βsym|2dx. (5.37)
Note that C >0 does not depend onk, rnorβ.
For every k= 1, . . . , Lr−1, the distance betweenWk andWk+1 can be estimated as follows
|Wk−Wk+1|2≤2
Urk∩Urk+1
|Wk−β|2+|Wk+1−β|2dx≤ 4C
|Urk∩Urk+1| ˆ
Urk∪Urk+1
|βsym|2dx.
x+εu(x) b=εe1
Figure 5.7: Sketch of the almost optimal displacement constructed in the proof of Lemma 5.A.3 for ξ=e1,ε= 14 andr=45.
Consequently, we obtain for1≤k < l≤Lr
|Wk−Wl|2=
l−1
X
i=k
Wi−Wi+1
2
≤(l−k)
l−1
X
i=k
|Wi−Wi+1|2
≤Lr Lr−1
X
i=1
4C
|Uri∩Uri+1| ˆ
Uri∪Uri+1
|βsym|2dx. (5.38)
We defineW =W1. Then, we derive from (5.37) and (5.38) the following chain of inequalities ˆ
B1(0)\Br(0)
|β−W|2≤2
Lr
X
k=1
ˆ
Urk
|β−Wk|2+|W −Wk|2dx
≤2
Lr
X
k=1
C ˆ
Urk
|βsym|2dx+Lr Lr−1
X
i=1
4C |Uri|
|Uri∩Uri+1| ˆ
Uri∪Uri+1
|βsym|2dx
!
≤4C ˆ
B1(0)\Br(0)
|βsym|2dx+ 16C L2r
Lr−1
X
i=1
ˆ
Uri∪Uri+1
|βsym|2dx
≤C(4 + 64L2r) ˆ
B1(0)\Br(0)
|βsym|2dx.
Note that by definitionL2r≤ (1−r)16π22 andC is a universal constant.
Finally, we construct for eachξ ∈R2 a function which is curl-free, whose circulation is exactlyξ, and whose elastic energy is optimal in scaling, see Figure 5.7.
Lemma 5.A.3. Let ξ ∈R2. There exists 0< r0 <1 such that for allr0 < r <1 andξ ∈R2 there exists a functionβ :B1(0)\Br(0)→R2×2 such thatcurlβ= 0 and´
∂B1β·τ dH1=ξand min
W∈Skew(2)
ˆ
B1(0)\Br(0)
|β−W|2dx≥c(1−r)−2 ˆ
B1(0)\Br(0)
|βsym|2dx. (5.39) The constantcdoes not depend on r, ξorβ.
Proof. Letξ= (ξ1, ξ2)∈R2. Moreover, we writeeρ(θ) = (cosθ,sinθ)andeθ(θ) = (−sinθ,cosθ).
We define the functionu:B1(0)\Br(0)→R2 in polar coordinates by
u(ρeρ(θ)) = 1 π
ˆ θ 0
(ξ1cosϕ+ξ2sinϕ)eρ(ϕ)dϕ+ (1−ρ)(ξ1cosθ+ξ2sinθ)eθ(θ)
! .
For a visualization, see Figure 5.7.
The function u has a jump on the line {θ = 0} of height ξ. Hence, the absolutely continuous part of the derivative of u, β =∇u, satisfies the circulation condition´
B1β·τ dH1=ξand curlβ = 0in B1(0)\Br(0). In addition, we can compute explicitly
β=1 π
1
ρ(ξ1cosθ+ξ2sinθ)eθ(θ)⊗eρ(θ)−(ξ1cosθ+ξ2sinθ)eρ(θ)⊗eθ(θ)
−1−ρ
ρ (ξ1cosθ+ξ2sinθ)eθ(θ)⊗eρ(θ) +1−ρ
ρ (−ξ1sinθ+ξ2cosθ)eθ(θ)⊗eθ(θ)
=1 π
−(ξ1cosθ+ξ2sinθ)eρ(θ)⊗eθ(θ) + (ξ1cosθ+ξ2sinθ)eθ(θ)⊗eρ(θ)
+ 1 π
1−ρ
ρ (−ξ1sinθ+ξ2cosθ)eθ(θ)⊗eθ(θ)
=βskew+βsym. One sees directly that ´
B1(0)\Br(0)βskewdx = 0. Hence, minW∈Skew(2)´
B1(0)\Br(0)|β −W|2dx =
´
B1(0)\Br(0)|β|2dx. In addition, a straightforward computation show that asr→1we obtain ˆ
B1(0)\Br(0)
|βskew|2dx= (1−r2)1
π|ξ|2∼2(1−r)1 π|ξ|2 and
ˆ
B1(0)\Br(0)
|βsym|2dx=
−log(r)−2(1−r) +1
2(1−r2) 1
π|ξ|2∼1
3(1−r)3 1 π|ξ|2, which proves the claim for rclose enough to1 independently ofξ.
Bibliography
[1] R. A. Adams and J. J. F. Fournier.Sobolev Spaces, volume 140 ofPure and Applied Mathematics.
Academic Press, Amsterdam, 2nd edition, 2003.
[2] R. Alicandro, M. Cicalese, and M. Ponsiglione. Variational equivalence between Ginzburg-Landau,XY spin systems and screw dislocations energies.Indiana Univ. Math. J., 60(1):171–208, 2011.
[3] R. Alicandro, L. De Luca, A. Garroni, and M. Ponsiglione. Metastability and dynamics of discrete topological singularities in two dimensions: aΓ-convergence approach.Arch. Ration. Mech. Anal., 214(1):269–330, 2014.
[4] R. Alicandro, L. De Luca, A. Garroni, and M. Ponsiglione. Dynamics of discrete screw dislocations on glide directions. J. Mech. Phys. Solids, 92:87–104, 2016.
[5] 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.
[6] M. Ariza and M. Ortiz. Discrete crystal elasticity and discrete dislocations in crystals. Arch.
Ration. Mech. Anal., 178(2):149–226, 2005.
[7] D. Bacon, D. Barnett, and R. O. Scattergood. Anisotropic continuum theory of lattice defects.
Prog. Mater. Sci., 23:51–262, 1980.
[8] J. Bassani. Incompatibility and a simple gradient theory of plasticity. J. Mech. Phys. Solids, 49(9):1983–1996, 2001.
[9] F. Béthuel, H. Brézis, and F. Hélein.Ginzburg-Landau Vortices. Progress in Nonlinear Differential Equations and their Applications, 13. Birkhäuser Boston, Inc., Boston, MA, 1994.
[10] J. Bourgain. On square functions on the trigonometric system. Bull. Soc. Math. Belg. Sér. B, 37:20–26, 1985.
[11] J. Bourgain and H. Brézis. On the equationdiv Y =f and application to control of phases. J.
Amer. Math. Soc., 16(2):393–426, 2003.
[12] J. Bourgain and H. Brézis. New estimates for elliptic equations and Hodge type systems. J.
European Math. Soc., 9(2):277–315, 2007.
[13] A. Braides.Γ-convergence for Beginners, volume 22 ofOxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002.
[14] H. Brézis and J. Van Schaftingen. Boundary estimates for elliptic systems withL1- data. Calc.
Var. Partial Differential Equations, 30:369–388, 2007.
[15] J. M. Burgers. Selected Papers of J. M. Burgers, chapter Physics. — Some considerations on the fields of stress connected with dislocations in a regular crystal lattice. I, pages 335–389. Springer Netherlands, Dordrecht, 1995.
[16] S. Cacace and A. Garroni. A multi-phase transition model for dislocations with interfacial mi-crostructure. Interfaces Free Bound., 11(2):291–316, 2009.
[17] P. Cermelli and G. Leoni. Renormalized energy and forces on dislocations. SIAM J. Math. Anal., 37(4):1131–1160, 2005.
[18] R. Coifman, J. L. Rubio de Francia, and S. Semmes. Multiplicateurs de Fourier de Lp(R) et estimations quadratiques. C. R. Acad. Sci. Paris Sér. I Math., 306(8):351–354, 1988.
[19] S. Conti, G. Dolzmann, and S. Müller. Korn’s second inequality and geometric rigidity with mixed growth conditions. Calc. Var. Partial Differential Equations, 50(1–2):437–454, 2014.
[20] S. Conti, A. Garroni, and A. Massaccesi. Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity. Calc. Var. Partial Differential Equations, 54(2):1847–1874, 2015.
[21] S. Conti, A. Garroni, and S. Müller. Singular kernels, multiscale decomposition of microstructure, and dislocation models. Arch. Ration. Mech. Anal., 199(3):779–819, 2011.
[22] S. Conti, A. Garroni, and S. Müller. Dislocation microstructures and strain-gradient plasticity with one active slip plane. http://arxiv.org/abs/1512.03076, 2015.
[23] S. Conti, A. Garroni, and M. Ortiz. The line-tension approximation as the dilute limit of linear-elastic dislocations. Arch. Ration. Mech. Anal., 218(2):699–755, 2015.
[24] S. Conti and P. Gladbach. A line-tension model of dislocation networks on several slip planes.
In P. Ariza, M. Ortiz, and V. Tvergaard, editors, Proceedings of the IUTAM Symposium on Micromechanics of Defects in Solids, volume 90, pages 140–147, 2015.
[25] S. Conti and M. Ortiz. Dislocation microstructures and the effective behavior of single crystals.
Arch. Ration. Mech. Anal., 176(1):103–147, 2005.
[26] C. M. Dafermos. Some remarks on Korn’s inequality. Z. Angew. Math. Phys., 19(6):913–920, 1968.
[27] G. Dal Maso. An Introduction to Γ-Convergence, volume 8 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 1993.
[28] G. Dal Maso, M. Negri, and D. Percivale. Linearized elasticity as Γ-limit of finite elasticity.
Set-Valued Anal., 10(2-3):165–183, 2002.
[29] R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology, volume 3. Springer, 1988.
[30] L. de Luca, A. Garroni, and M. Ponsiglione. Γ-convergence analysis of systems of edge disloca-tions: the self energy regime. Arch. Ration. Mech. Anal., 206(3):885–910, 2012.
[31] G. Dolzmann, N. Hungerbühler, and S. Müller. Uniqueness and maximal regularity for nonlinear elliptic systems ofn-Laplace type with measure valued right hand side. J. Reine Angew. Math., 520:1–35, 2000.
[32] L. Evans and R. Gariepy.Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.
[33] N. Fleck and J. Hutchinson. A phenomenological theory for strain gradient effects in plasticity.
J. Mech. Phys. Solids, 41(12):1825–1857, 1993.
[34] N. Fleck and J. Hutchinson. A reformulation of strain gradient plasticity. J. Mech. Phys. Solids, 49(10):2245–2271, 2001.
[35] J. Frenkel. Zur Theorie der Elastizitätsgrenze und der Festigkeit kristallinischer Körper.Z. Phys., 37(7-8):572–609, 1926.
[36] K. Friedrichs. On the boundary-value problems of the theory of elasticity and Korn’s inequality.
Ann. of Math., 48(2):441–471, 1947.
[37] 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(11):1461–
1506, 2002.
[38] A. Garroni, G. Leoni, and M. Poniglione. Gradient theory for plasticity via homogenization of discrete dislocations. J. Eur. Math. Soc., 12(5):1231–1266, 2010.
[39] A. Garroni and S. Müller. Γ-limit of a phase-field model of dislocations. SIAM J. Math. Anal., 36(6):1943–1964, 2005.
[40] A. Garroni and S. Müller. A variational model for dislocations in the line tension limit. Arch.
Ration. Mech. Anal., 181(3):535–578, 2006.
[41] T. A. Gillespie and J. L. Torrea. Transference of a Littlewood-Paley-Rubio inequality and di-mension free estimates. Rev. Un. Mat. Argentina, 45(1):1–6, 2004.
[42] P. Gladbach. PhD thesis, University of Bonn, in preparation.
[43] L. Grafakos.Classical Fourier Analysis, volume 249 ofGraduate Texts in Mathematics. Springer, New York, 2nd edition, 2008.
[44] L. Grafakos, L. Liu, and D. Yang. Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math. Scand., 104(2):296–310, 2009.
[45] M. E. Gurtin. An Introduction to Continuum Mechanics, volume 158 ofMathematics in Science and Engineering. Academic Press, New York-London, 1981.
[46] M. E. Gurtin. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids, 50(1):5–32, 2002.
[47] J. P. Hirth and J. Lothe. Theory of Dislocations. McGraw-Hill, New York, 1968.
[48] C. O. Horgan. Korn’s inequalities and their applications in continuum mechanics. SIAM Rev., 37(4):491–511, 1995.
[49] C. O. Horgan and L. E. Payne. On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch.
Ration. Mech. Anal., 82(2):165–179, 1983.
[50] D. Hull and D. J. Bacon. Introduction to Dislocations. Butterworth-Heinemann, Oxford, UK, 5th edition, 2011.
[51] R. L. Jerrard. Lower bounds for generalized Ginzburg-Landau functionals. J. Math. Anal., 30(4):721–746, 1999.
[52] V. Kondrat’ev and O. Oleˇinik. Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities. Uspekhi Math. Nauk, 43(5(263)):55–98, 1988.
[53] A. Korn. Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen. Bull. Intern. Cracov. Akad. umiejet (Classe Sci. Math. Nat.), 13:706–724, 1909.
[54] M. Koslowski, A. M. Cuitino, and M. Ortiz. A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals.J. Mech. Phys. Solids, 50(12):2597–2635, 2002.
[55] M. T. Lacey. Issues related to Rubio de Francia’s Littlewood-Paley inequality, volume 2 ofNYJM Monographs. State University of New York, University at Albany, Albany, NY, 2007.
[56] S. Luckhaus and L. Mugnai. On a mesoscopic many-body hamiltonian describing elastic shears and dislocations. Contin. Mech. Thermodyn., 22(4):251–290, 2010.
[57] V. Maz’ya. Bourgain-Brézis type inequality with explicit constants. Contemp. Math., 445:247–
252, 2007.
[58] S. Müller and M. Palombaro. Derivation of rod theory for biphase materials with dislocations at the interface. Calc. Var. Partial Differential Equations, 48(3-4):315–335, 2013.
[59] S. Müller, L. Scardia, and C. I. Zeppieri. Geometric rigidity for incompatible fields and an application to strain-gradient plasticity. Indiana Univ. Math. J., 63(5):1365–1396, 2014.
[60] S. Müller, L. Scardia, and C. I. Zeppieri. Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations. InAnalysis and Computation of Microstructure in Finite Plasticity, volume 78 ofLect. Notes Appl. Comput. Mech., pages 175–204. Springer, 2015.
[61] F. R. N. Nabarro. Dislocations in a simple cubic lattice. Proc. Phys. Soc., 59(2):256–272, 1947.
[62] W. D. Nix and H. Gao. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids, 46(3):411–425, 1998.
[63] J. Nye. Some geometrical relations in dislocated crystals. Acta Metall., 1(2):153–162, 1953.
[64] L. Payne and H. Weinberger. On Korn’s inequality. Arch. Ration. Mech. Anal., 8(1):89–98, 1961.
[65] R. Peierls. The size of a dislocation. Proc. Phys. Soc., 52(1):34–37, 1940.
[66] M. Polanyi. Über eine Art Gitterstörung, die einen Kristall plastisch machen könnte. Z. Phys., 89(9-10):660–664, 1934.
[67] M. Polanyi and E. Schmid. Zur Frage der Plastizität. Verformung bei tiefen Temperaturen.
Naturwissenschaften, 17(18-19):301–304, 1929.