The equilibrium measure for a nonlocal dislocation energy. (English) Zbl 1412.49004
The paper is devoted to studying the equilibrium measure for a nonlocal and anisotropic weighted energy \(I\) describing the interaction of positive dislocations in the plane. It is proved that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, some measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In the paper, a positive answer to the conjecture that positive dislocations tend to form vertical walls is given. To find explicitly the unique minimizer of the nonlocal energy \(I\), the following approach is used: the strict convexity of the nonlocal energy \(I\) on the class of measure with compact support and finite interaction energy is proved. Strict convexity implies uniqueness of the minimizer and the equivalence between minimality and the Euler-Lagrange conditions for \(I\). Then, it is shown that the semicircle law satisfies the Euler-Lagrange conditions and hence is the unique minimizer of \(I\).
Reviewer: Aygul Manapova (Ufa)
MSC:
| 49J10 | Existence theories for free problems in two or more independent variables |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
References:
| [1] | Alicandro, R.; Cicalese, M.; Ponsiglione, M.Variational equivalence between Ginzburg‐Landau, XY spin systems and screw dislocations energies. Indiana Univ. Math. J.60 (2011), no. 1, 171-208. doi: 10.1512/iumj.2011.60.4339 · Zbl 1251.49017 |
| [2] | Alicandro, R.; Ponsiglione, M.Ginzburg‐Landau functionals and renormalized energy: a revised Γ‐convergence approach. J. Funct. Anal.266 (2014), no. 8, 4890-4907. doi: 10.1016/j.jfa.2014.01.024 · Zbl 1307.35287 |
| [3] | Balagué, D.; Carrillo, J. A.; Laurent, T.; Raoul, G.Dimensionality of local minimizers of the interaction energy. Arch. Ration. Mech. Anal.209 (2013), no. 3, 1055-1088. doi: 10.1007/s00205-013-0644-6 · Zbl 1311.49053 |
| [4] | Baskaran, R.; Akarapu, S.; Mesarovic, S. D.; Zbib, H. M.Energies and distributions of dislocations in stacked pile‐ups. Int. J. Solids Struc.47 (2010), no. 9, 1144-1153. doi: 10.1016/j.ijsolstr.2010.01.007 · Zbl 1193.74110 |
| [5] | Berdichevsky, V. L.On dislocation models of grain boundary. Contin. Mech. Thermodyn.23 (2011), no. 3, 185-209. doi: 10.1007/s00161-010-0173-6 · Zbl 1272.74025 |
| [6] | Cañizo, J. A.; Carrillo, J. A.; Patacchini, F. S.Existence of compactly supported global minimisers for the interaction energy. Arch. Ration. Mech. Anal.217 (2015), no. 3, 1197-1217. doi: 10.1007/s00205-015-0852-3 · Zbl 1317.82010 |
| [7] | Carrillo, J. A.; Castorina, D.; Volzone, B.Ground states for diffusion dominated free energies with logarithmic interaction. SIAM J. Math. Anal.47 (2015), no. 1, 1-25. doi: 10.1137/140951588 · Zbl 1323.35095 |
| [8] | Carrillo, J. A.; Delgadino, M. G.; Mellet, A.Regularity of local minimizers of the interaction energy via obstacle problems. Comm. Math. Phys.343 (2016), no. 3, 747-781. doi: 10.1007/s00220-016-2598-7 · Zbl 1337.49066 |
| [9] | Carrillo, J. A.; DiFrancesco, M.; Figalli, A.; Laurent, T.; Slepčev, D.Confinement in nonlocal interaction equations. Nonlinear Anal.75 (2012), no. 2, 550-558. doi:10.1016/j.na.2011.08.057 · Zbl 1233.35032 |
| [10] | Carrillo, J. A.; Hittmeir, S.; Volzone, B.; Yao, Y.Nonlinear aggregation‐diffusion equations: radial symmetry and long time asymptotics. Preprint, 2016. arxiv:1603.07767 [math.AP] · Zbl 1427.35136 |
| [11] | Duda, F. P.; šilhavý, M.Dislocation walls in crystals under single slip. Comput. Methods Appl. Mech. Engrg.193 (2004), no. 48-51, 5385-5409. doi: 10.1016/j.cma.2003.12.069 · Zbl 1112.74351 |
| [12] | Folland, G. B.Introduction to partial differential equations. Second edition. Princeton University Press, Princeton, N.J., 1995. · Zbl 0841.35001 |
| [13] | Frostman, O.Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel. Lunds Univ. Mat. Sem.3 (1935), 1-118. · JFM 61.1262.02 |
| [14] | Geers, M. G. D.; Peerlings, R. H. J.; Peletier, M. A.; Scardia, L.Asymptotic behaviour of a pile‐up of infinite walls of edge dislocations. Arch. Ration. Mech. Anal.209 (2013), no. 2, 495-539. doi: 10.1007/s00205-013-0635-7 · Zbl 1282.74067 |
| [15] | Hall, C. L.Asymptotic analysis of a pile‐up of regular edge dislocation walls. Mat. Sci. Eng.: A530 (2011), 144-148. doi: 10.1016/j.msea.2011.09.065 |
| [16] | Hirth, J. P.; Lothe, J.Theory of dislocations. Wiley, New York, 1982. |
| [17] | Lubarda, V. A.; Blume, J. A.; Needleman, A.An analysis of equilibrium dislocation distributions. Acta Metall. Mater.41 (1993), no. 2, 625-642. doi: 10.1016/0956-7151(93)90092-7 |
| [18] | Mehta, M. L.Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. · Zbl 1107.15019 |
| [19] | Mora, M. G.; Peletier, M. A.; Scardia, L.Convergence of interaction‐driven evolutions of dislocations with Wasserstein dissipation and slip‐plane confinement. SIAM J. Math. Anal.49 (2017), no. 5, 4149-4205. doi: 10.1137/16M1096098 · Zbl 1378.49012 |
| [20] | Rudin, W.Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw‐Hill, New York, 1991. · Zbl 0867.46001 |
| [21] | Saada, G.; Bouchaud, E.Dislocation walls. Acta Metall. Mater.41 (1993), no. 7, 2173-2178. doi: 10.1016/0956-7151(93)90387-8 |
| [22] | Saff, E. B.; Totik, V.Logarithmic potentials with external fields. Grundlehren der mathematischen Wissenschaften, 316. Springer, Berlin, 1997. doi: 10.1007/978-3-662-03329-6 · Zbl 0881.31001 |
| [23] | Sandier, E.; Serfaty, S.1D log gases and the renormalized energy: crystallization at vanishing temperature. Probab. Theory Related Fields162 (2015), no. 3‐4, 795-846. doi: 10.1007/s00440-014-0585-5 · Zbl 1327.82005 |
| [24] | Sandier, E.; Serfaty, S.2D Coulomb gases and the renormalized energy. Ann. Probab.43 (2015), no. 4, 2026-2083. doi: 10.1214/14-AOP927 · Zbl 1328.82006 |
| [25] | Simione, R.; Slepčev, D.; Topaloglu, I.Existence of ground states of nonlocal‐interaction energies. J. Stat. Phys.159 (2015), no. 4, 972-986. doi:10.1007/s10955-015-1215-z · Zbl 1328.82016 |
| [26] | vanMeurs, P.; Muntean, A.Upscaling of the dynamics of dislocation walls. Adv. Math. Sci. Appl.24 (2014), no. 2, 401-414. · Zbl 1326.74110 |
| [27] | Wigner, E. P.Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2)62 (1955), 548-564. doi: 10.2307/1970079 · Zbl 0067.08403 |
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