Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
By X. Cabré, M. Fall, J. Solà-Morales, Tobias Weth. Journal für die Reine und Angewandte Mathematik, 2015
71 citations recorded by the source corpus.
Abstract We are concerned with hypersurfaces of ℝ N {\mathbb{R}^{N}} with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ℝ N {\mathbb{R}^{N}} with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ℝ 2 {\mathbb{R}^{2}} with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/0624f4c22b2178f4cbf6376a928aeaecdbba68fe
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In Applied Mathematics
- C. Bucur
- Lenka Slav'ikov'a
- L. Pick
- Andrea Cianchi
- Enrico Valdinoci
- A. Alberico
- Vicentiu D. Rădulescu
- Giovanni Molica Bisci
In Computational Theory and Mathematics
- C. Bucur
- Andrea Cianchi
- Enrico Valdinoci
- A. Alberico
- L. Pick
- Vicentiu D. Rădulescu
- Giovanni Molica Bisci
- Raffaella Servadei
Citations in the corpus
- Nonlocal Diffusion and Applications (2015, background): “See also [14] for the construction of surfaces with constant mean curvature with the structure of onduloids, and [14] and [29] for the proof that bounded surfaces with constant mean curvature are necessarily spheres (this is the analogue of a celebrated result by Alexandrov for surfaces of constant ”
- Fractional Orlicz-Sobolev embeddings (2020, background): “A touch of recent contributions in this connection is furnished by [7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 41, 44, 45, 51, 52, 53, 58, 59, 61, 62, 63, 64, 65, 68, 70, 71, 77, 76, 78, 83, 86].”