publication . Article . Preprint . 2012

Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces

Begoña Barrios; Alessio Figalli; Enrico Valdinoci;
Open Access
  • Published: 21 Feb 2012
We prove that $C^{1,\alpha}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.
free text keywords: Mathematics - Analysis of PDEs
Related Organizations
Funded by
NSF| Analytical and geometrical problems in calculus of variations and partial differential equations
  • Funder: National Science Foundation (NSF)
  • Project Code: 0969962
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences
Elliptic Pdes and Symmetry of Interfaces and Layers for Odd Nonlinearities
  • Funder: European Commission (EC)
  • Project Code: 277749
  • Funding stream: FP7 | SP2 | ERC
19 references, page 1 of 2

[1] L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math. 134 (2011), no. 3-4, 377-403.

[2] C. Bjorland, L. Caffarelli and A. Figalli, Non-Local Tug-of-War and the Infinity Fractional Laplacian, Comm. Pure Applied Math., to appear.

[3] C. Bjorland, L. Caffarelli and A. Figalli, Non-Local Gradient Dependent Operators, preprint.

[4] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111-1144.

[5] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62 (2009), no. 5, 597-638

[6] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Rational Mech. Anal. 200 (2011), no. 1, 59-88.

[7] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations, preprint,

[8] L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 1-23.

[9] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41 (2011), no. 1-2, 203-240.

[10] L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, preprint,

[11] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, preprint,

[12] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer, Berlin (2001).

[13] O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, preprint,

[14] O. Savin and E. Valdinoci, Γ-convergence for nonlocal phase transitions, preprint,

[15] O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, preprint, arc-bin/mpa?yn=12-8

19 references, page 1 of 2
Any information missing or wrong?Report an Issue