Minimization of a fractional perimeter-Dirichlet integral functional

Caffarelli, Luis; Savin, Ovidiu; Valdinoci, Enrico

doi:10.1016/j.anihpc.2014.04.004

Caffarelli, Luis ; Savin, Ovidiu ; Valdinoci, Enrico

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 901-924.

Résumé

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely

\int_{Ω} {| \nabla u (x) |}^{2} d x + {Per}_{σ} ({u > 0}, Ω),

with

σ \in (0, 1)

. We obtain regularity results for the minimizers and for their free boundaries

\partial {u > 0}

using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2014.04.004

Mots clés : Free boundary problems, Fractional minimal surfaces, Regularity theory

@article{AIHPC_2015__32_4_901_0,
     author = {Caffarelli, Luis and Savin, Ovidiu and Valdinoci, Enrico},
     title = {Minimization of a fractional {perimeter-Dirichlet} integral functional},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {901--924},
     publisher = {Elsevier},
     volume = {32},
     number = {4},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.04.004},
     mrnumber = {3390089},
     zbl = {1323.35216},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.04.004/}
}

TY  - JOUR
AU  - Caffarelli, Luis
AU  - Savin, Ovidiu
AU  - Valdinoci, Enrico
TI  - Minimization of a fractional perimeter-Dirichlet integral functional
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 901
EP  - 924
VL  - 32
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2014.04.004/
DO  - 10.1016/j.anihpc.2014.04.004
LA  - en
ID  - AIHPC_2015__32_4_901_0
ER  -

%0 Journal Article
%A Caffarelli, Luis
%A Savin, Ovidiu
%A Valdinoci, Enrico
%T Minimization of a fractional perimeter-Dirichlet integral functional
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 901-924
%V 32
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.04.004/
%R 10.1016/j.anihpc.2014.04.004
%G en
%F AIHPC_2015__32_4_901_0

Caffarelli, Luis; Savin, Ovidiu; Valdinoci, Enrico. Minimization of a fractional perimeter-Dirichlet integral functional. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 901-924. doi : 10.1016/j.anihpc.2014.04.004. http://www.numdam.org/articles/10.1016/j.anihpc.2014.04.004/

Bibliographie
Cité par

[1] H.W. Alt, L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105 -144 | EuDML | MR | Zbl

[2] H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, Trans. Am. Math. Soc. 282 no. 2 (1984), 431 -461 | MR | Zbl

[3] L. Ambrosio, G. De Philippis, L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscr. Math. 134 no. 3–4 (2011), 377 -403 | MR | Zbl

[4] I. Athanasopoulos, L.A. Caffarelli, C. Kenig, S. Salsa, An area-Dirichlet integral minimization problem, Commun. Pure Appl. Math. 54 no. 4 (2001), 479 -499 | MR | Zbl

[5] B. Barrios, A. Figalli, E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) (2014), http://dx.doi.org/10.2422/2036-2145.201202_007 | MR | Zbl

[6] L. Caffarelli, J.-M. Roquejoffre, O. Savin, Nonlocal minimal surfaces, Commun. Pure Appl. Math. 63 no. 9 (2010), 1111 -1144 | MR | Zbl

[7] L. Caffarelli, E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differ. Equ. 41 no. 1–2 (2011), 203 -240 | MR | Zbl

[8] C.-K. Chen, P.C. Fife, Nonlocal models of phase transitions in solids, Adv. Math. Sci. Appl. 10 (2000), 821 -849 | MR | Zbl

[9] S. Dipierro, A. Figalli, G. Palatucci, E. Valdinoci, Asymptotics of the s-perimeter as $s ↘ 0$ , Discrete Contin. Dyn. Syst. 33 no. 7 (2013), 2777 -2790 | MR | Zbl

[10] V. Maz'Ya, T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 no. 2 (2002), 230 -238 | MR | Zbl

[11] O. Savin, E. Valdinoci, Density estimates for a nonlocal variational model via the Sobolev inequality, SIAM J. Math. Anal. 43 no. 6 (2011), 2675 -2687 | MR | Zbl

[12] O. Savin, E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differ. Equ. 48 no. 1–2 (2013), 33 -39 | MR | Zbl

[13] O. Savin, E. Valdinoci, Some monotonicity results for minimizers in the calculus of variations, J. Funct. Anal. 264 no. 10 (2013), 2469 -2496 | MR | Zbl

Cité par Sources :