O. Alvarez, E. Carlini, R. Monneau, and E. Rouy, Convergence of a first order scheme for a non-local eikonal equation, Applied Numerical Mathematics, vol.56, issue.9, pp.1136-1146, 2006.
DOI : 10.1016/j.apnum.2006.03.002

O. Alvarez, P. Hoch, Y. L. Bouar, and R. Monneau, R??solution en temps court d'une ??quation de Hamilton???Jacobi non??locale d??crivant la dynamique d'une dislocation, Comptes Rendus Mathematique, vol.338, issue.9, pp.679-684, 2004.
DOI : 10.1016/j.crma.2004.03.007

O. Alvarez, P. Hoch, Y. L. Bouar, and R. Monneau, Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution, Archive for Rational Mechanics and Analysis, vol.51, issue.3, pp.449-504, 2006.
DOI : 10.1007/s00205-006-0418-5

G. Barles, Solutions de Viscosité des Equations de Hamilton-Jacobi, 1994.

J. D. Benamou and P. Hoch, GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type, ESAIM: Mathematical Modelling and Numerical Analysis, vol.36, issue.5, pp.883-905, 2002.
DOI : 10.1051/m2an:2002037
URL : https://hal.archives-ouvertes.fr/inria-00072179

W. Bollmann, Interference Effects in the Electron Microscopy of Thin Crystal Foils, Physical Review, vol.103, issue.5, pp.1588-1589, 1956.
DOI : 10.1103/PhysRev.103.1588

L. Cheng, D. J. Srolovitz, E. Weinan, and Y. Xiang, A level set method for dislocation dynamics, Acta Mater, vol.51, pp.5499-5518, 2003.

M. G. Crandall and P. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, vol.43, issue.167, pp.1-19, 1984.
DOI : 10.1090/S0025-5718-1984-0744921-8

A. M. Cuitiño, M. Koslowski, and M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, J. Mech. Phys. Solids, vol.50, issue.12, pp.2597-2635, 2002.

A. Garroni and S. Müller, $\Gamma$-Limit of a Phase-Field Model of Dislocations, SIAM Journal on Mathematical Analysis, vol.36, issue.6, 1943.
DOI : 10.1137/S003614100343768X

A. Ghorbel, M. Rhabi, and R. Monneau, Comportement mécanique par homogénéisation de la dynamique des disclocations, Proceedings du Colloque National MECAMAT -Aussois, 2006.

A. Ghorbel, P. Hoch, and R. Monneau, A numerical study for the homogenization of one-dimensional models describing the motion of dislocations, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00110528

A. Ghorbel and R. Monneau, Equation d'Hamilton-Jacobi non-locale modélisant la dynamique des dislocations, Proceedings of the 2nd TAM-TAM (Trends in Applied Mathematics in Tunisia, pp.322-328, 2005.

A. Ghorbel and R. Monneau, Well-posedness of a non-local transport equation modelling dislocations dynamics , preprint Cermics-ENPC 304, 2006.

P. B. Hirsch, R. W. Horne, and M. S. Whelan, LXVIII. Direct observations of the arrangement and motion of dislocations in aluminium, Philosophical Magazine, vol.43, issue.7, pp.677-684, 1956.
DOI : 10.1088/0370-1301/64/9/301

J. R. Hirth and L. Lothe, Theory of dislocations, Second Edition, 1992.

C. Imbert, R. Monneau, and E. Rouy, Homogenization of first order equations with (u/?)-periodic Hamiltonians. Part II, preprint on HAL server of CNRS, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00016270

P. Lions, G. Papanicolaou, and S. R. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished preprint, 1986.

F. R. Nabarro, Theory of crystal dislocations, 1969.

E. Orowan, Z. Kristallplastizität, I. , and Z. Phys, Zur Kristallplastizit???t. I, 22] M. Polanyi, ¨ Uber eine Art Gitterstörung, die einem Kristall plastisch machen könnte, pp.605-634, 1934.
DOI : 10.1007/BF01341478

D. Rodney, Y. L. Bouar, and A. Finel, Phase field methods and dislocations, Acta Materialia, vol.51, issue.1, pp.17-30, 2003.
DOI : 10.1016/S1359-6454(01)00379-2
URL : https://hal.archives-ouvertes.fr/hal-00125516

G. I. Taylor, The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.145, issue.855, pp.362-387, 1934.
DOI : 10.1098/rspa.1934.0106

D. J. Bacon and D. Hull, Introduction to Dislocations, 1984.

A. H. Cottrell and A. H. , Dislocations and Plastic Flow in Crystals, American Journal of Physics, vol.22, issue.4, 1953.
DOI : 10.1119/1.1933704

J. R. Hirth and L. Lothe, Theory of dislocations, Second Edition, 1992.

R. W. Lardner, Mathematical theory of dislocations and fracture, Mathematical Expositions, vol.17, 1974.

F. R. Nabarro, Theory of crystal dislocations, 1969.

O. Alvarez, P. Hoch, Y. Le-bouar, and R. Monneau, R??solution en temps court d'une ??quation de Hamilton???Jacobi non??locale d??crivant la dynamique d'une dislocation, Comptes Rendus Mathematique, vol.338, issue.9, pp.679-684, 2004.
DOI : 10.1016/j.crma.2004.03.007

O. Alvarez, P. Hoch, Y. Le-bouar, and R. Monneau, Dislocation dynamics driven by the self-force : short time existence and uniqueness of the solution

O. Alvarez, E. Carlini, P. Hoch, Y. Le-bouar, and R. Monneau, Dislocation dynamics described by non-local Hamilton???Jacobi equations, Materials Science and Engineering: A, vol.400, issue.401
DOI : 10.1016/j.msea.2005.01.062

G. Barles, Solutions de viscosité deséquationsdeséquations de Hamilton-Jacobi, 1994.

J. R. Hirth and L. Lothe, Theory of dislocations, p.Krieger, 1992.

D. Rodney, Y. L. Bouar, and A. , Phase field methods and dislocations, Acta Materialia, vol.51, issue.1, pp.17-30, 2003.
DOI : 10.1016/S1359-6454(01)00379-2
URL : https://hal.archives-ouvertes.fr/hal-00125516

E. Rouy and A. Tourin, A Viscosity Solutions Approach to Shape-From-Shading, References [1] O. Alvarez, P. Cardaliaguet and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces and Free Boundaries, pp.867-884, 1992.
DOI : 10.1137/0729053

O. Alvarez, E. Carlini, R. Monneau, and E. Rouy, Convergence of a first order scheme for a non-local eikonal equation, Applied Numerical Mathematics, vol.56, issue.9
DOI : 10.1016/j.apnum.2006.03.002

O. Alvarez, E. Carlini, R. Monneau, and E. Rouy, A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics, Numerische Mathematik, vol.51, issue.18
DOI : 10.1007/s00211-006-0030-5

M. Bardi and I. Capuzzo-dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, 1997.
DOI : 10.1007/978-0-8176-4755-1

G. Barles, Solutions de Viscosité des Equations de Hamilton-Jacobi, 1994.

G. Barles, P. Cardaliaguet, O. Ley, and R. Monneau, General results for dislocation type equation

G. Barles and O. Ley, Nonlocal First-Order Hamilton???Jacobi Equations Modelling Dislocations Dynamics, Communications in Partial Differential Equations, vol.6, issue.8
DOI : 10.1007/BFb0094298
URL : https://hal.archives-ouvertes.fr/hal-00021694

M. G. Crandall, H. Ishii, and P. Lions, user's guide to viscosity solutions\\ of second order\\ partial differential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.1-67, 1992.
DOI : 10.1090/S0273-0979-1992-00266-5

M. G. Crandall and P. Lions, Conditions d'unicité pour les solutions généralisées deséquationsdeséquations de Hamilton-Jacobi du premier ordre, C. R. Acad

M. G. Crandall and P. Lions, On existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlinear Anal, pp.353-370, 1986.

M. G. Crandall and P. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, vol.43, issue.167, pp.1-19, 1984.
DOI : 10.1090/S0025-5718-1984-0744921-8

F. Da-lio, N. Forcadel, and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics, Journal of the European Mathematical Society
DOI : 10.4171/JEMS/140
URL : https://hal.archives-ouvertes.fr/hal-00415939

A. Ghorbel, P. Hoch, and R. Monneau, A One Dimensional Numerical Homogeneization for Several Dislocations

A. Ghorbel and R. Monneau, Equation d'Hamilton-Jacobi non-locale modélisant la dynamique des dislocations, Proceedings of the 2nd TAM-TAM (Trends in Applied Mathematics in Tunisia, pp.322-328, 2005.

J. R. Hirth and L. Lothe, Theory of dislocations, Second Edition, 1992.

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, pp.12-49, 1988.
DOI : 10.1016/0021-9991(88)90002-2

G. Barles, Solutions de Viscosité des Equations de Hamilton-Jacobi, 1994.

M. Bardi and I. Capuzzo-dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, 1997.
DOI : 10.1007/978-0-8176-4755-1

E. Rouy and A. Tourin, A Viscosity Solutions Approach to Shape-From-Shading, SIAM Journal on Numerical Analysis, vol.29, issue.3, pp.867-884, 1992.
DOI : 10.1137/0729053

A. Ghorbel and R. Monneau, ´ Equation d'Hamilton-Jacobi non locale modélisant la dynamique des dislocations, Proceedings of the 2 nd TAM-TAM, (Trends in Applied Mathematics in Tunisia, pp.322-328, 2005.

A. Ghorbel and R. Monneau, Well-posedeness of a non-local transport equation modelling dislocations dynamics

J. P. Hirth and J. Lothe, Theory of dislocations, 1982.

E. Rouy and A. Tourin, A Viscosity Solutions Approach to Shape-From-Shading, SIAM Journal on Numerical Analysis, vol.29, issue.3, pp.867-884, 1992.
DOI : 10.1137/0729053

]. R. References1 and . Abgrall, Numerical discretization of boundary conditions for first order Hamilton-Jacobi equations, SIAM J. Numer. Anal, vol.41, issue.6, pp.2233-2261, 2003.

M. Bardi and I. Capuzzo-dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, 1997.
DOI : 10.1007/978-0-8176-4755-1

G. Barles, Solutions de Viscosité des Equations de Hamilton-Jacobi, 1994.

M. G. Crandall and P. Lions, Conditions d'unicité pour les solutions généralisées deséquationsdeséquations de Hamilton-Jacobi du premier ordre, C. R. Acad

A. Ghorbel, M. Rhabi, and R. Monneau, Comportement mécanique par homogénéisation de la dynamique des disclocations, Proceedings of Colloque National MECAMAT -Aussois 2006, 2006.

A. Ghorbel and R. Monneau, Well-posedness of a non-local transport equation modelling dislocations dynamics , preprint Cermics-ENPC 304, 2006.

D. A. Gomes and A. M. Oberman, Computing the Effective Hamiltonian Using a Variational Approach, SIAM Journal on Control and Optimization, vol.43, issue.3, pp.792-812, 2004.
DOI : 10.1137/S0363012902417620

J. R. Hirth and L. Lothe, Theory of dislocations, Second Edition, 1992.

J. Kratochvil, R. Sela?ek, and E. Werner, The importance of being curved : bowing dislocations in a continuum description, Philos. Mag, vol.83, pp.31-34, 2003.

P. Lions, G. Papanicolaou, and S. R. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished preprint, 1986.

M. Rorro, An approximation scheme for the effective Hamiltonian and applications, Applied Numerical Mathematics, vol.56, issue.9, pp.1238-1254, 2006.
DOI : 10.1016/j.apnum.2006.03.006

B. Générale and [. Abgrall, Numerical discretization of boundary conditions for first order Hamilton-Jacobi equations, SIAM J. Numer. Anal, vol.41, issue.6, pp.2233-2261, 2003.

P. [. Alvarez, R. Cardaliaguet, and . Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces and Free Boundaries, pp.415-434, 2005.

E. [. Alvarez, P. Carlini, Y. L. Hoch, R. Bouar, and . Monneau, Dislocation dynamics described by non-local Hamilton???Jacobi equations, Proceedings in Materials Science & Engineering, pp.400-401, 2005.
DOI : 10.1016/j.msea.2005.01.062

E. [. Alvarez, R. Carlini, E. Monneau, and . Rouy, A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics, Numerische Mathematik, vol.51, issue.18, pp.413-444, 2006.
DOI : 10.1007/s00211-006-0030-5

P. [. Alvarez, Y. L. Hoch, R. Bouar, and . Monneau, R??solution en temps court d'une ??quation de Hamilton???Jacobi non??locale d??crivant la dynamique d'une dislocation, Comptes Rendus Mathematique, vol.338, issue.9, pp.679-684, 2004.
DOI : 10.1016/j.crma.2004.03.007

P. [. Alvarez, Y. L. Hoch, R. Bouar, and . Monneau, Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution, Archive for Rational Mechanics and Analysis, vol.51, issue.3, pp.449-504, 2006.
DOI : 10.1007/s00205-006-0418-5

M. Bardi and I. Capuzzo-dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, 1997.
DOI : 10.1007/978-0-8176-4755-1

G. Barles, Solutions de viscosité deséquationsdeséquations de Hamilton-Jacobi, 1994.

P. [. Barles, O. Cardaliaguet, R. Ley, and . Monneau, General results for dislocation type equation

O. [. Barles and . Ley, Nonlocal First-Order Hamilton???Jacobi Equations Modelling Dislocations Dynamics, Communications in Partial Differential Equations, vol.6, issue.8, pp.1191-1208, 2006.
DOI : 10.1007/BFb0094298
URL : https://hal.archives-ouvertes.fr/hal-00021694

W. Bollmann, Interference Effects in the Electron Microscopy of Thin Crystal Foils, Physical Review, vol.103, issue.5, pp.1588-1589, 1956.
DOI : 10.1103/PhysRev.103.1588

[. Cheng, D. J. Srolovitz, E. Weinan, and Y. Xiang, A level set method for dislocation dynamics, Acta Mater, vol.51, pp.5499-5518, 2003.

A. H. Cottrell, Dislocations and Plastic Flow in Crystals, American Journal of Physics, vol.22, issue.4, 1953.
DOI : 10.1119/1.1933704

M. G. Crandall and P. Lions, Conditions d'unicité pour les solutions généralisées deséquationsdeséquations de Hamilton-Jacobi du premier ordre, C. R. Acad

M. G. Crandall and P. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, vol.43, issue.167, pp.1-19, 1984.
DOI : 10.1090/S0025-5718-1984-0744921-8

M. G. Crandall and P. L. Lions, On existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlinear Anal, pp.353-370, 1986.

M. G. Crandall, H. Ishii, and P. L. Lions, user's guide to viscosity solutions\\ of second order\\ partial differential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.1-67, 1992.
DOI : 10.1090/S0273-0979-1992-00266-5

M. [. Cuitiño, M. Koslowski, and . Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, J. Mech. Phys. Solids, vol.50, issue.12, pp.2597-2635, 2002.

R. [. Elliott, B. E. Schätzle, and . Stoth, Viscosity Solutions of a Degenerate Parabolic-Elliptic System Arising in the Mean-Field Theory of Superconductivity, Archive for Rational Mechanics and Analysis, vol.145, issue.2, pp.99-127, 1998.
DOI : 10.1007/s002050050125

A. Garroni and S. Müller, $\Gamma$-Limit of a Phase-Field Model of Dislocations, SIAM Journal on Mathematical Analysis, vol.36, issue.6, 1943.
DOI : 10.1137/S003614100343768X

M. [. Ghorbel, R. Rhabi, and . Monneau, Comportement mécanique par homogénéisation de la dynamique des disclocations, Proceedings du Colloque National MECAMAT -Aussois, 2006.

P. [. Ghorbel, R. Hoch, and . Monneau, A numerical study for the homogenization of one-dimensional models describing the motion of dislocations, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00110528

R. [. Ghorbel and . Monneau, ´ Equation d'Hamilton-Jacobi non-locale modélisant la dynamique des dislocations, Proceedings of the 2nd TAM-TAM (Trends in Applied Mathematics in Tunisia, pp.322-328, 2005.

R. [. Ghorbel and . Monneau, Well-posedness of a non-local transport equation modelling dislocations dynamics , preprint Cermics-ENPC 304, 2006.

A. [. Gomes and . Oberman, Computing the Effective Hamiltonian Using a Variational Approach, SIAM Journal on Control and Optimization, vol.43, issue.3, pp.792-812, 2004.
DOI : 10.1137/S0363012902417620

R. [. Hirsch, M. J. Horne, and . Whelan, Direct Observations of the Arrangement and Motion of Dislocations in Aluminium Phil, Mag, vol.1, pp.677-684, 1956.

J. R. Hirth and L. Lothe, Theory of dislocations, Second Edition, 1992.

R. [. Imbert, E. Monneau, and . Rouy, Homogenization of first order equations with (u/?)-periodic Hamiltonians. Part II, preprint on HAL server of CNRS, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00016270

. [. Ishii, Existence and uniqueness of solutions of Hamilton-Jacobi equations, Funkcial. Ekvac, vol.29, pp.167-188, 1986.

J. Kratochvil, R. Sela?ek, and E. Werner, The importance of being curved : bowing dislocations in a continuum description, Philos. Mag, vol.83, pp.31-34, 2003.

. [. Lardner, Mathematical theory of dislocations and fracture, Mathematical Expositions, vol.17, 1974.

[. Lions, G. Papanicolaou, and S. R. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished preprint, 1986.

J. [. Osher and . Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, pp.12-49, 1988.
DOI : 10.1016/0021-9991(88)90002-2

. [. Peierls, The size of a dislocation, Proceedings of the Physical Society, vol.52, issue.1, pp.34-37, 1940.
DOI : 10.1088/0959-5309/52/1/305

. [. Polanyi, A. Uber, and . Gitterstörung, ???ber eine Art Gitterst???rung, die einen Kristall plastisch machen k???nnte, Zeitschrift f???r Physik, vol.89, issue.9-10, p.660, 1934.
DOI : 10.1007/BF01341481

Y. [. Rodney, A. Bouar, and . Finel, Phase field methods and dislocations, Phase field methods and dislocations, pp.17-30, 2003.
DOI : 10.1016/S1359-6454(01)00379-2
URL : https://hal.archives-ouvertes.fr/hal-00125516

M. Rorro, An approximation scheme for the effective Hamiltonian and applications, Applied Numerical Mathematics, vol.56, issue.9, pp.1238-1254, 2006.
DOI : 10.1016/j.apnum.2006.03.006

E. Rouy and A. Tourin, A Viscosity Solutions Approach to Shape-From-Shading, SIAM Journal on Numerical Analysis, vol.29, issue.3, pp.867-884, 1992.
DOI : 10.1137/0729053

. [. Taylor, The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.145, issue.855, pp.362-387, 1934.
DOI : 10.1098/rspa.1934.0106