Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries

Journal of the American Mathematical Society

Volumn 21, Issue 3, 2008, Pages 847-862

  Caffarelli, L A ^a   Lin, Fang Hua ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Harmonic maps; Multiple valued harmonic functions; Regularity of free interface; Singular limit

Indexed keywords

EID: 58149361424     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-08-00593-6     Document Type: Article

References (23)

1. Almgren, F. J., Jr. Q valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two. Bull. Amer. Math. Soc. (N. S.) 8 (1983), no. 2, 327-328. MR684900 (84b:49052)
2. Bucur, D.; Buttazzo, G.; Henrot, A. Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8 (1998), no. 2, 571-579. MR1657219 (99k:49004)
3. Burke, S. P.; Schuman, T. E. W. Diffusion flames. Industr. Eng. Chem. 20 (1928), no. 10, 998-1004.
4. Buttazzo, G.; Dal Maso, G. Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991), no. 1, 17-49. MR1076053 (92e:49055) (Pubitemid 21696010)
5. Buttazzo, G.; Dal Maso, G. An existence result for a class ofshape optimization problems. Arch. Rational Mech. Anal. 122 (1993), no. 2, 183-195. MR1217590 (94i:49052)
6. Bucur, D.; Zokésio, J.-P. N-dimensional shape optimization under capacitary constraint. J. Differential Equations 123 (1995), no. 2, 504-522. MR1362884 (97g:49054)
7. Caffarelli, L. A.; Lin, F. An optimal partition problem for eigenvalues. J. Sci. Comput. 31 (2007), no. 1, 5-18. Available online at: http://www.springerlink.com/content/v671661832u5/?p= d2c1296d08c347ee80e9718c28995186&pi=4. MR2304268 (Pubitemid 46643750)
8. Caffarelli, L. A.; Lin, F. Viscosity approach to a partition problem, preprint.
9. Chang, S.-M.; Lin, C.-S.; Lin, T.-C.; Lin, W.-W. Segregated nodal domains of twodimensional multispecies Bose-Einstein condensates. Phys. D 196 (2004), nos. 3-4, 341-361. MR2090357 (2005g:82074)
10. Caffarelli, L. A.; Roquejoffre, J.-M. Uniform Hölder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames. Arch. Ration. Mech. Anal. 183 (2007), no. 3, 457-487. MR2278412
11. Conti, M.; Terracini, S.; Verzini, G. Nehari's problem and competing species systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 6, 871-888. MR1939088 (2003i:35084)
12. Conti, M.; Terracini, S.; Verzini, G. An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198 (2003), no. 1, 160-196. MR1962357 (2004h:35171) (Pubitemid 36374135)
13. Conti, M.; Terracini, S.; Verzini, G. Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195 (2005), no. 2, 524-560. MR2146353 (2006b:35077) (Pubitemid 40664845)
14. Conti, M.; Terracini, S.; Verzini, G. A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. 54 (2005), no. 3, 779-815. MR2151234 (2006b:35180) (Pubitemid 41095838)
15. p weights and unique continuation. Indiana Univ. Math. J. 35 (1986), no. 2, 245-268. MR833393 (88b:35059)
16. Jerison, D. S.; Kenig, C. E. Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. in Math. 46 (1982), no. 1, 80-147. MR676988 (84d:31005b)
17. Lin, F.-H. Static and moving vortices in Ginzburg-Landau theories. Nonlinear partial differential equations in geometry and physics (Knoxville, TN, 1995), 71-111. Progr. Nonlinear Differential Equations Appl., 29. Birkhäuser, Basel, 1997. MR1437152 (97m:35241)
18. Lin, F.; Yang, X. Geometric measure theory-an introduction. Advanced Mathematics (Beijing/Boston), 1. Science Press, Beijing; International Press, Boston, 2002. MR2030862 (2005a:28001)
19. Lin, F.-H. Nodal sets of solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 44 (1991), no. 3, 287-308. MR1090434 (92b:58224)
20. Morrey, C. B., Jr. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, 130. Springer, New York, 1966. MR0202511 (34:2380)
21. Schoen, R. M. Analytic aspects of the harmonic map problem. Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), 321-358. Math. Sci. Res. Inst. Publ., 2. Springer, New York, 1984. MR765241 (86b:58032)
22. Šverák, V. On optimal shape design. J. Math. Pures Appl. (9) 72 (1993), no. 6, 537-551. MR1249408 (94j:49047)
23. Ziemer, W. P. Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989. MR1014685 (91e:46046)