Density estimates for a variational model driven by the Gagliardo norm

Journal des Mathematiques Pures et Appliquees

Volumn 101, Issue 1, 2014, Pages 1-26

  Savin, Ovidiu ^a   Valdinoci, Enrico ^b,c  

^a Och Spine at New York Presbyterian Hospitals   (United States)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^c UNIVERSITY OF MILAN   (Italy)

Author keywords

Gagliardo norm; Minimizers; Variational model

Indexed keywords

EID: 84888440147     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2013.05.001     Document Type: Article

References (28)

1. Alberti G., Bouchitté G., Seppecher P. Phase transition with the line-tension effect. Arch. Ration. Mech. Anal. 1998, 144(1):1-46.
2. L. Ambrosio, G. de Philippis, L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, preprint, 2010.
3. Bouchitté G. Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 1990, 21(3):289-314.
4. s,p when s↑1 and applications. J. Anal. Math. 2002, 87:77-101. (Dedicated to the memory of Thomas H. Wolff).
5. Brezis H., Mironescu P. Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. J. Evol. Equ. 2001, 1(4):387-404.
6. Cabré X., Solà-Morales J. Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 2005, 58(12):1678-1732.
7. Cabré X., Cinti E. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete Contin. Dyn. Syst. 2010, 28(3):1179-1206.
8. Caffarelli L.A., Córdoba A. Uniform convergence of a singular perturbation problem. Comm. Pure Appl. Math. 1995, 48(1):1-12.
9. Caffarelli L.A., Roquejoffre J.-M., Savin O. Nonlocal minimal surfaces. Comm. Pure Appl. Math. 2010, 63(9):1111-1144.
10. Caffarelli L.A., Souganidis P.E. Convergence of nonlocal threshold dynamics approximations to front propagation. Arch. Ration. Mech. Anal. 2010, 195(1):1-23.
11. Caffarelli L.A., Valdinoci E. Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differential Equations 2011, 41(1-2):203-240.
12. Cinti E. Stime dell'energia e simmetria 1-dimensionale per equazioni frazionarie 2009, Mathematical analysis seminar, Università di Bologna. http://www.dm.unibo.it/seminario-pini/.
13. Evans L.C. Partial Differential Equations. Grad. Stud. Math. 1998, vol. 19. American Mathematical Society, Providence, RI, xviii+662 pp.
14. Farina A., Valdinoci E. Geometry of quasiminimal phase transitions. Calc. Var. Partial Differential Equations 2008, 33(1):1-35.
15. Gagliardo E. Un criterio di compattezza rispetto alla convergenza in media. Ric. Mat. 1957, 6:34-48.
16. Gagliardo E. Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 1957, 27:284-305.
17. Garroni A., Palatucci G. A singular perturbation result with a fractional norm. Progr. Nonlinear Differential Equations Appl. 2006, vol. 68:111-126. Birkhäuser, Basel.
18. González M.d.M. Gamma convergence of an energy functional related to the fractional Laplacian. Calc. Var. Partial Differential Equations 2009, 36(2):173-210.
19. Modica L. The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 1987, 98(2):123-142.
20. --convergenza. Boll. Un. Mat. Ital. B (5) 1977, 14(1):285-299.
21. Owen N.C., Sternberg P. Nonconvex variational problems with anisotropic perturbations. Nonlinear Anal. 1991, 16(7-8):705-719.
22. Palatucci G., Savin O., Valdinoci E. Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. Pura Appl. (4) 2012, http://www.springerlink.com/content/g37m26q02843107k/, 10.1007/s10231-011-0243-9.
23. Savin O. Regularity of flat level sets in phase transitions. Ann. of Math. (2) 2009, 169(1):41-78.
24. Savin O., Valdinoci E. Γ-convergence for nonlocal phase transitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 2012, 29(4):479-500.
25. Savin O., Valdinoci E. Density estimates for a nonlocal variational model via the Sobolev inequality. SIAM J. Math. Anal. 2011, 43(6):2675-2687.
26. Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator 2005, Ph.D. thesis, University of Texas at Austin. http://www.math.uchicago.edu/~luis/preprints.html.
27. Silvestre L. Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 2006, 55(3):1155-1174.
28. Sire Y., Valdinoci E. Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 2009, 256(6):1842-1864.