Existence and uniqueness for dislocation dynamics with nonnegative velocity

Interfaces and Free Boundaries

Volumn 7, Issue 4, 2005, Pages 415-434

  Alvarez, O ^a   Cardaliaguet, P ^b   Monneau, R ^c  

^a UNIVERSITÉ DE ROUEN   (France)

^b UNIVERSITÉ DE BRETAGNE OCCIDENTALE   (France)

^c UNIVERSITÉ PARIS EST   (France)

Author keywords

Discontinuous viscosity solutions; Dislocation dynamics; Eikonal equation; Hamilton Jacobi equations; Nonlocal equations

Indexed keywords

EID: 30644456591     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/131     Document Type: Article

References (21)

1. ALIBAUD, N. Existence and uniqueness for non-linear parabolic equations with non-local terms. Preprint (2004).
2. ALVAREZ, O., HOCH, P., LE BOUAR, Y., & MONNEAU, R. Résolution en temps court d'une équation de Hamilton-Jacobi non locale décrivant la dynamique d'une dislocation. C. R. Acad. Sci. Parts Sér. 1338 (2004), 679-684. Zbl 1049.35005 MR 2065373
3. ALVAREZ, O., HOCH, P., LE BOUAR, Y., & MONNEAU, R. Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rat. Mech. Anal., to appear.
4. AMBROSIO, L., FUSCO, N., & PALLARA, D., Functions of Bounded Variation and Free Discontinuity Problem. Oxford Math. Monographs, Clarendon Press, Oxford (2000). Zbl 0957.49001 MR 1857292
5. BARDI, M. & CAPUZZO-DOLCETTA, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). Zbl 0890.49011 MR 1484411
6. BARLES, G. Solutions de Viscosité des Equations de Hamilton-Jacobi. Springer, Berlin (1994). Zbl 0819.35002 MR 1613876
7. BARLES, G., SONER, H. M., & SOUGANIDIS, P. E. Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993), 439-469. Zbl 0785.35049 MR 1205984
8. CANNARSA, P. & CARDALIAGUET, P. Perimeter estimates for the reachable set of control problems. J. Convex Analysis, to appear.
9. CANNARSA, P. & FRANKOWSKA, H. Interior sphere property of attainable sets and time optimal control. Preprint (2004).
10. CANNARSA, P. & SINESTRARI, C. Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhäuser, Boston (2004). Zbl pre02129788 MR 2041617
11. CARDALIAGUET, P., On front propagation problems with nonlocal terms. Adv. Differential Equations 5 (2000), 213-268. Zbl 1029.53081 MR 1734542
12. CAROFF, N. & FRANKOWSKA, H. Conjugate points and shocks in nonlinear optimal control. Trans. Amer. Math. Soc. 348 (1996), 3133-3153. Zbl 0924.49016 MR 1344204
13. CLARKE, F. The maximum principle under minimal hypotheses. SIAM J. Control and Optim. 14 (1976), 1078-1091. Zbl 0344.49009 MR 0415453
14. EVANS, L. C. & GARIEPY, R. F., Measure Theory and Fine Properties of Functions. Stud. Adv. Math., CRC Press, Boca Raton, FL (1992). Zbl 0804.28001 MR 1158660
15. FEDERER, H. Geometric Measure Theory, Springer (1969). Zbl 0176.00801 MR 0257325
16. HIRTH, J. R. & LOTHE, L. Theory of Dislocations. 2nd ed., Krieger, Malabar, FL (1992).
17. KUBIN, L. P., CANOVA, G., CONDAT, M., DEVINCRE, B., PONTIKIS, V., & BRÉCHET, Y. Dislocation microstructures and plastic flow: a 3D simulation. Solid State Phenomena 23 & 24 (1992), 455.
18. LARDNER, R. W. Mathematical Theory of Dislocations and Fracture. Mathematical Expositions 17, Univ. of Toronto Press (1974).
19. LORENZ, T. Boundary regularity of reachable sets of control systems. Systems Control Lett. 54 (2005), 919-924.
20. NABARRO, F. R. N. Theory of Crystal Dislocations. Clarendon Press, Oxford (1969).
21. RODNEY, D., LE BOUAR, Y., & FINEL, A. Phase field methods and dislocations, Acta Mater. 51 (2003), 17-30.