Comptes Rendus
Mathematical analysis/Functional analysis
A note on the fractional perimeter and interpolation
[Une note sur le périmètre fractionnaire et l'interpolation]
Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 960-965.

Nous présentons le périmètre fractionnaire en tant que fonction d'ensemble qui interpole la mesure de Lebesgue et le périmètre au sens de De Giorgi. Notre motivation provient d'une inégalité fractionnaire que nous avons récemment démontrée dans l'esprit de la Boxing inequality de W. Gustin reliant le périmètre fractionnaire et le contenu de Hausdorff. Cette nouvelle inégalité permet de retrouver des propriétés de la semi-norme de Gagliardo dans le cadre des espaces de Sobolev Wα,1 d'ordre 0<α<1.

We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces Wα,1 of order 0<α<1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.09.001

Augusto C. Ponce 1 ; Daniel Spector 2, 3

1 Université catholique de Louvain, Institut de recherche en mathématique et physique, chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium
2 National Chiao Tung University, Department of Applied Mathematics, Hsinchu, Taiwan
3 National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei 106, Taiwan
@article{CRMATH_2017__355_9_960_0,
     author = {Augusto C. Ponce and Daniel Spector},
     title = {A note on the fractional perimeter and interpolation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {960--965},
     publisher = {Elsevier},
     volume = {355},
     number = {9},
     year = {2017},
     doi = {10.1016/j.crma.2017.09.001},
     language = {en},
}
TY  - JOUR
AU  - Augusto C. Ponce
AU  - Daniel Spector
TI  - A note on the fractional perimeter and interpolation
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 960
EP  - 965
VL  - 355
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2017.09.001
LA  - en
ID  - CRMATH_2017__355_9_960_0
ER  - 
%0 Journal Article
%A Augusto C. Ponce
%A Daniel Spector
%T A note on the fractional perimeter and interpolation
%J Comptes Rendus. Mathématique
%D 2017
%P 960-965
%V 355
%N 9
%I Elsevier
%R 10.1016/j.crma.2017.09.001
%G en
%F CRMATH_2017__355_9_960_0
Augusto C. Ponce; Daniel Spector. A note on the fractional perimeter and interpolation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 960-965. doi : 10.1016/j.crma.2017.09.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.001/

[1] C. Bennett; R. Sharpley Interpolation of Operators, Pure and Applied Mathematics, vol. 129, Academic Press, Boston, MA, USA, 1988

[2] J. Bourgain; H. Brezis; P. Mironescu Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001, pp. 439-455

[3] L. Brasco; E. Lindgren; E. Parini The fractional Cheeger problem, Interfaces Free Bound., Volume 16 (2014), pp. 419-458

[4] M. Cwikel Monotonicity properties of interpolation spaces, Ark. Mat., Volume 14 (1976), pp. 213-236

[5] J. Dávila On an open question about functions of bounded variation, Calc. Var. Partial Differ. Equ., Volume 15 (2002), pp. 519-527

[6] W. Gustin Boxing inequalities, J. Math. Mech., Volume 9 (1960), pp. 229-239

[7] V.I. Kolyada; A.K. Lerner On limiting embeddings of Besov spaces, Stud. Math., Volume 171 (2005), pp. 1-13

[8] V.G. Maz'ya; T.O. Shaposhnikova On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002), pp. 230-238 (Erratum: J. Funct. Anal., 201, 2003, pp. 298-300)

[9] M. Milman Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 3425-3442

[10] A.C. Ponce Elliptic PDEs, Measures and Capacities. From the Poisson Equation to Nonlinear Thomas–Fermi Problems, EMS Tracts in Mathematics, vol. 23, European Mathematical Society (EMS), Zürich, Switzerland, 2016

[11] A.C. Ponce; D. Spector A boxing inequality for the fractional perimeter (submitted for publication) | arXiv

Cité par Sources :

Commentaires - Politique