Short time existence of the classical solution to the fractional mean curvature flow

Julin, Vesa; La Manna, Domenico Angelo

doi:10.1016/j.anihpc.2020.02.007

Julin, Vesa ; La Manna, Domenico Angelo

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 983-1016.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and $C^{1, 1}$ -regular. We provide the same result also for the volume preserving fractional mean curvature flow.

MR Zbl

DOI : 10.1016/j.anihpc.2020.02.007

Mots-clés : Fractional mean curvature flow, Short time existence, Classical solution, Fractional perimeter

@article{AIHPC_2020__37_4_983_0,
     author = {Julin, Vesa and La Manna, Domenico Angelo},
     title = {Short time existence of the classical solution to the fractional mean curvature flow},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {983--1016},
     publisher = {Elsevier},
     volume = {37},
     number = {4},
     year = {2020},
     doi = {10.1016/j.anihpc.2020.02.007},
     mrnumber = {4104832},
     zbl = {1501.53096},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2020.02.007/}
}

TY  - JOUR
AU  - Julin, Vesa
AU  - La Manna, Domenico Angelo
TI  - Short time existence of the classical solution to the fractional mean curvature flow
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 983
EP  - 1016
VL  - 37
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2020.02.007/
DO  - 10.1016/j.anihpc.2020.02.007
LA  - en
ID  - AIHPC_2020__37_4_983_0
ER  -

%0 Journal Article
%A Julin, Vesa
%A La Manna, Domenico Angelo
%T Short time existence of the classical solution to the fractional mean curvature flow
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 983-1016
%V 37
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2020.02.007/
%R 10.1016/j.anihpc.2020.02.007
%G en
%F AIHPC_2020__37_4_983_0

Julin, Vesa; La Manna, Domenico Angelo. Short time existence of the classical solution to the fractional mean curvature flow. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 983-1016. doi : 10.1016/j.anihpc.2020.02.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.02.007/

Bibliographie
Cité par

[1] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000 | DOI | MR | Zbl

[2] Aubin, T. Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998 | DOI | MR | Zbl

[3] Barrios, B.; Figalli, A.; Valdinoci, E. Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 13 (2014), pp. 609–639 | MR | Zbl

[4] Cabré, X.; Fall, M.M.; Solá-Morales, J.; Weth, T. Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay, J. Reine Angew. Math., Volume 745 (2018), pp. 253–280 | MR | Zbl

[5] Caffarelli, L.; Roquejoffre, J.-M.; Savin, O. Nonlocal minimal surfaces, Commun. Pure Appl. Math., Volume 63 (2010), pp. 1111–1144 | DOI | MR | Zbl

[6] Caffarelli, L.; Souganidis, P.E. Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., Volume 195 (2010), pp. 1–23 | DOI | MR | Zbl

[7] Cesaroni, A.; Dipierro, S.; Novaga, M.; Valdinoci, E. Fattening and nonfattening phenomena for planar nonlocal curvature flows, Math. Ann., Volume 375 (2019), pp. 687–736 | DOI | MR | Zbl

[8] Chambolle, A.; Morini, M.; Ponsiglione, M. A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal., Volume 44 (2012), pp. 4048–4077 | DOI | MR | Zbl

[9] Chambolle, A.; Morini, M.; Ponsiglione, M. Nonlocal curvature flows, Arch. Ration. Mech. Anal., Volume 218 (2015), pp. 1263–1329 | DOI | MR | Zbl

[10] Chambolle, A.; Novaga, M.; Ruffini, B. Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound., Volume 19 (2017), pp. 393–415 | DOI | MR | Zbl

[11] Cinti, E.; Sinestrari, C.; Valdinoci, E. Neckpinch singularities in fractional mean curvature flows, Proc. Am. Math. Soc., Volume 146 (2018), pp. 2637–2646 | DOI | MR | Zbl

[12] E. Cinti, C. Sinestrari, E. Valdinoci, Convex sets evolving by volume preserving fractional mean curvature flows, preprint, 2018. | MR

[13] Ciraolo, G.; Figalli, A.; Maggi, F.; Novaga, M. Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature, J. Reine Angew. Math., Volume 741 (2018), pp. 275–294 | MR | Zbl

[14] Dávila, J.; del Pino, M.; Wei, J. Nonlocal $s$ -minimal surfaces and Lawson cones, J. Differ. Geom., Volume 109 (2018), pp. 111–175 | DOI | MR | Zbl

[15] Evans, L.C. Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998 | MR | Zbl

[16] Evans, L.C.; Spruck, J. Motion of level sets by mean curvature. II, Trans. Am. Math. Soc., Volume 330 (1992), pp. 321–332 | DOI | MR | Zbl

[17] Figalli, A.; Fusco, N.; Maggi, F.; Millot, V.; Morini, M. Isoperimetry and stability properties of balls with respect to nonlocal energies, Commun. Math. Phys., Volume 336 (2015), pp. 441–507 | DOI | MR | Zbl

[18] Frank, R.L.; Seiringer, R. Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., Volume 255 (2008), pp. 3407–3430 | DOI | MR | Zbl

[19] Fusco, N.; Millot, V.; Morini, M. A quantitative isoperimetric inequality for fractional perimeters, J. Funct. Anal., Volume 261 (2011), pp. 697–715 | DOI | MR | Zbl

[20] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001 | DOI | MR | Zbl

[21] Huisken, G. Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., Volume 20 (1984), pp. 237–266 | DOI | MR | Zbl

[22] Huisken, G. The volume preserving mean curvature flow, J. Reine Angew. Math., Volume 382 (1987), pp. 35–48 | MR | Zbl

[23] Huisken, G.; Polden, A. Calculus of Variations and Geometric Evolution Problems, Geometric evolution equations for hypersurfaces, Springer-Verlag, Berlin (1999), pp. 45–84 (Cetraro, 1996) | MR | Zbl

[24] Imbert, C. Level set approach for fractional mean curvature flows, Interfaces Free Bound., Volume 11 (2009), pp. 153–176 | DOI | MR | Zbl

[25] Lee, J.M. Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997 | DOI | MR | Zbl

[26] Mantegazza, C. Lecture Notes on Mean Curvature Flow, Progress in Mathematics, vol. 290, Birkhäuser/Springer, Basel, 2011 | DOI | MR | Zbl

[27] Mikulevicius, R.; Pragarauskas, H. On Hölder solutions of the integro-differential Zakai equation, Stoch. Process. Appl., Volume 119 (2009), pp. 3319–3355 | DOI | MR | Zbl

[28] Sáez, M.; Valdinoci, E. On the evolution by fractional mean curvature, Commun. Anal. Geom., Volume 27 (2019) | DOI | MR | Zbl

[29] Savin, O.; Valdinoci, E. Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differ. Equ., Volume 48 (2013), pp. 33–39 | DOI | MR | Zbl

Cité par Sources :