Geodesic
A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 1, pp. 185-214

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The classical Michael–Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, thanks to an additional L p term on the right-hand side which is weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo seminorm of the function, as well as its L p norm weighted by the fractional mean curvature of the hypersurface.

As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.

Accepté le :
Publié le :
DOI : 10.4171/aihpc/39
Classification : 26D10, 46E35, 52A20, 53A07
Keywords: Fractional Sobolev inequalities on manifolds, nonlocal mean curvature, fractional mean curvature flow, maximal time of existence, convexity 
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     title = {A fractional {Michael{\textendash}Simon} {Sobolev} inequality on convex hypersurfaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {185--214},
     volume = {40},
     number = {1},
     year = {2023},
     doi = {10.4171/aihpc/39},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/39/}
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Cabré, Xavier; Cozzi, Matteo; Csató, Gyula. A fractional Michael–Simon Sobolev inequality on convex hypersurfaces. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 1, pp. 185-214. doi: 10.4171/aihpc/39

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