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 of order .
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 d'ordre .
Accepted:
Published online:
Augusto C. Ponce 1; Daniel Spector 2, 3
@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}, }
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] Interpolation of Operators, Pure and Applied Mathematics, vol. 129, Academic Press, Boston, MA, USA, 1988
[2] Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001, pp. 439-455
[3] The fractional Cheeger problem, Interfaces Free Bound., Volume 16 (2014), pp. 419-458
[4] Monotonicity properties of interpolation spaces, Ark. Mat., Volume 14 (1976), pp. 213-236
[5] On an open question about functions of bounded variation, Calc. Var. Partial Differ. Equ., Volume 15 (2002), pp. 519-527
[6] Boxing inequalities, J. Math. Mech., Volume 9 (1960), pp. 229-239
[7] On limiting embeddings of Besov spaces, Stud. Math., Volume 171 (2005), pp. 1-13
[8] 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] Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 3425-3442
[10] 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 boxing inequality for the fractional perimeter (submitted for publication) | arXiv
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