Minimizing movements for dislocation dynamics with a mean curvature term

Forcadel, Nicolas; Monteillet, Aurélien

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 214-244.

We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.

Classification : 53C44, 49Q15, 49L25, 28A75, 58A25
Mots clés : front propagation, non-local equations, dislocation dynamics, mean curvature motion, viscosity solutions, minimizing movements, sets of finite perimeter, currents

[1] F. Almgren, J.E. Taylor and L. Wang, Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387-438. | MR | Zbl

[2] O. Alvarez, P. Cardaliaguet and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Boundaries 7 (2005) 415-434. | MR | Zbl

[3] O. Alvarez, E. Carlini, R. Monneau and E. Rouy, A convergent scheme for a nonlocal Hamilton-Jacobi equation, modeling dislocation dynamics. Num. Math. 104 (2006) 413-572. | MR | Zbl

[4] O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rational Mech. Anal. 85 (2006) 371-414. | MR | Zbl

[5] L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR | Zbl

[6] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005). | MR | Zbl

[7] G. Barles, P. Cardaliaguet, O. Ley and R. Monneau, Global existence results and uniqueness for dislocation type equations. SIAM J. Math. Anal. (to appear). | MR | Zbl

[8] G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32 (1995) 484-500. | MR | Zbl

[9] G. Barles and O. Ley, Nonlocal first-order hamilton-jacobi equations modelling dislocations dynamics. Comm. Partial Differential Equations 31 (2006) 1191-1208. | MR | Zbl

[10] G. Barles, H.M. Soner and P.E. Souganidis, Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439-469. | MR | Zbl

[11] E. Bombieri, Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78 (1982) 99-130. | MR | Zbl

[12] P. Cardaliaguet, On front propagation problems with nonlocal terms. Adv. Differential Equations 5 (2000) 213-268. | MR | Zbl

[13] P. Cardaliaguet and O. Ley, On the energy of a flow arising in shape optimisation. Interfaces Free Bound. (to appear). | MR | Zbl

[14] P. Cardaliaguet and D. Pasquignon, On the approximation of front propagation problems with nonlocal terms. ESAIM: M2AN 35 (2001) 437-462. | Numdam | MR | Zbl

[15] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR | Zbl

[16] L.C. Evans and J. Spruck, Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992) 321-332. | MR | Zbl

[17] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969). | MR | Zbl

[18] N. Forcadel, Dislocations dynamics with a mean curvature term: short time existence and uniqueness. Differential Integral Equations 21 (2008) 285-304. | MR

[19] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105. Princeton University Press, Princeton, NJ (1983). | MR | Zbl

[20] Y. Giga and S. Goto, Geometric evolution of phase-boundaries, in On the evolution of phase boundaries (Minneapolis, MN, 1990-1991), IMA Vol. Math. Appl. 43, Springer, New York (1992) 51-65. | MR | Zbl

[21] E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80. Birkhäuser Verlag, Basel (1984). | MR | Zbl

[22] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differential Equations 3 (1995) 253-271. | MR | Zbl

[23] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16. Birkhäuser Verlag, Basel (1995). | MR | Zbl

[24] Y. Maekawa, On a free boundary problem of viscous incompressible flows. Interfaces Free Bound. 9 (2007) 549-589. | MR | Zbl

[25] F. Morgan, Geometric measure theory. A beginner's guide. Academic Press Inc., Boston, MA (1988). | MR | Zbl

[26] R. Schoen, L. Simon and F.J. Almgren, Jr., Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II. Acta Math. 139 (1977) 217-265. | MR | Zbl

[27] L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Vol. 3, Australian National University Centre for Mathematical Analysis, Canberra (1983). | MR | Zbl

[28] P. Soravia and P.E. Souganidis, Phase-field theory for FitzHugh-Nagumo-type systems. SIAM J. Math. Anal. 27 (1996) 1341-1359. | MR | Zbl