A Fractional Framework for Perimeters and Phase Transitions

Enrico Valdinoci1
1Universita» degli Studi di Milano

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Tài liệu tham khảo

Ambrosio L., De Philippis G., Martinazzi L.: Γ-convergence of nonlocal perimeter functionals. Manuscripta Math. 134, 377–403 (2011)

B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integrodifferential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) http://arxiv.org/abs/1202.4606v1

Cabré X., Cinti E.: Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete Contin. Dyn. Syst. 28(3), 1179–1206 (2010)

X. Cabré, E. Cinti, Fractional diffusion equations: energy estimates and 1-D symmetry in dimension 3, preprint.

X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, preprint arxiv.org/abs/1111.0796

Cabré X., Solà-Morales J.: Layer solutions in a half-space for boundary reactions. Comm. Pure Appl. Math. 58(12), 1678–1732 (2005)

Caffarelli L., Córdoba A.: Uniform convergence of a singular perturbation problem. Comm. Pure Appl. Math. 48, 1–12 (1995)

Caffarelli L., Roquejoffre J.-M., Savin O.: Non-local minimal surfaces. Comm. Pure Appl. Math. 63, 1111–1144 (2010)

Caffarelli L., Valdinoci E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differential Equations 41(1-2), 203–240 (2011)

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, preprint http://arxiv.org/abs/1105.1158

E. De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Bologna, Pitagora (1979).

del Pino M., Kowalczyk M., Wei J.: A counterexample to a conjecture by De Giorgi in large dimensions. C. R. Acad. Sci. Paris, Ser. I 346, 1261–1266 (2008)

Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as $${s \searrow 0}$$ , Discrete Contin. Dyn. Syst. 33 (7) (2013) 2777–2790.

A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ., Hackensack, NJ (2009).

Imbert C.: Level set approach for fractional mean curvature flows. Interfaces Free Bound. 11, 153–176 (2009)

Mazya V. , Shaposhnikova T.: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195, 230–238 (2002)

Modica L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98(2), 123–142 (1987)

G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4), DOI: 10.1007/s10231-011-0243-9 http://dx.doi.org/10.1007/s10231-011-0243-9

O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, preprint http://arxiv.org/abs/1007.2114

Savin O., Valdinoci E.: Density estimates for a nonlocal variational model via the Sobolev inequality. SIAM J. Math. Analysis 43(6), 2675–2687 (2011)

Savin O., Valdinoci E.: Γ-convergence for nonlocal phase transitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4), 479–500 (2012)

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, DOI: 10.1007/s00526-012-0539-7 http://www.springerlink.com/content/467n313161531332

O. Savin and E. Valdinoci, Some monotonicity results for minimizers in the calculus of variations, J. Funct. Anal.

L. Silvestre, Regularity of the obstacle problem for a fractional power of the laplace operator, PhD thesis, University of Texas at Austin (2005) http://math.uchicago.edu/luis/preprints/luisdissreadable.pdf

Sire Y., Valdinoci E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256(6), 1842–1864 (2009)