Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows

References

• Mantegazza, C. (2011). Lecture notes on mean curvature flow. Prog. Math., 290. Birkhäuser/Springer Basel AG, Basel,
   Google Scholar
• Cesaroni, A., Dipierro, S., Novaga, M., Valdinoci, E. (2019). Fattening and nonfattening phenomena for planar nonlocal curvature flows. Math. Ann. 375(1–2):687–736. DOI: 10.1007/s00208-018-1793-6.
   Web of Science ®Google Scholar
• Chambolle, A., Novaga, M., Ruffini, B. (2017). Some results on anisotropic fractional mean curvature flows. Interfaces Free Bound. 19(3):393–415. DOI: 10.4171/IFB/387.
   Web of Science ®Google Scholar
• Imbert, C. (2009). Level set approach for fractional mean curvature flows. Interfaces Free Bound. 11(1):153–176.
   Web of Science ®Google Scholar
• Julin, V., La Manna, D. (2019). Short time existence of the classical solution to the fractional mean curvature flow. Ann. Inst. H. Poincaré Anal. non lin. 37(4):983–1016. DOI: 10.1016/j.anihpc.2020.02.007.
   Google Scholar
• Barles, G., Cardaliaguet, P., Ley, O., Monneau, R. (2008). Global existence results and uniqueness for dislocation equations. SIAM J. Math. Anal. 40(1):44–69. DOI: 10.1137/070682083.
   Web of Science ®Google Scholar
• Da Lio, F., Forcadel, N. (2008). R. Monneau: Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics. J. Eur. Math. Soc. (JEMS). 10(4):1061–1104.
   Web of Science ®Google Scholar
• Chen, Y.G., Giga, Y., Goto, S. (1991). Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33(3):749–786. DOI: 10.4310/jdg/1214446564.
   Web of Science ®Google Scholar
• Evans, L.C., Spruck, J. (1991). Motion of level sets by mean curvature I. J. Differential Geom. 33(3):635–681. DOI: 10.4310/jdg/1214446559.
   Web of Science ®Google Scholar
• Osher, S., Sethian, J.A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1):12–49. DOI: 10.1016/0021-9991(88)90002-2.
   Web of Science ®Google Scholar
• Chambolle, A., Morini, M., Ponsiglione, M. (2015). Nonlocal curvature flows. Arch. Rational Mech. Anal. . 218(3):1263–1329. DOI: 10.1007/s00205-015-0880-z.
   Web of Science ®Google Scholar
• Dipierro, S., Figalli, A., Palatucci, G., Valdinoci, E. (2013). Asymptotics of the s-perimeter as s↘0. Discrete Cont. Dynam. Syst. 33(7):2777–2790.
   Web of Science ®Google Scholar
• Maz’ya, V., Shaposhnikova, T. (2002). On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195(2):230–238.
   Web of Science ®Google Scholar
• De Luca, L., Novaga, M., Ponsiglione, M. The 0-fractional perimeter between fractional perimeters and Riesz potentials. To appear in Ann. SNS, arXiv: 1906.06303.
   Google Scholar
• Cesaroni, A., Novaga, M. (2020). Second order asymptotics of the fractional perimeter as s→1. Math. Eng. 2(3):527–556.
   Google Scholar
• Ambrosio, L., De Philippis, G., Martinazzi, L. (2011). Gamma-convergence of nonlocal perimeter functionals. Manuscripta Math. 134(3–4):377–403. DOI: 10.1007/s00229-010-0399-4.
   Web of Science ®Google Scholar
• Barchiesi, M., Kang, S. H., Le, T. M., Morini, M., Ponsiglione, M. (2010). A variational model for infinite perimeter segmentations based on Lipschitz level set functions: denoising while keeping finely oscillatory boundaries. Multiscale Model. Simul. 8(5):1715–1741. DOI: 10.1137/090773659.
   Web of Science ®Google Scholar
• Zhao, Y., Rada, L., Chen, K., Harding, S.P., Zheng, Y. (2015). Automated vessel segmentation using infinite perimeter active contour model with hybrid region information with application to retinal images. IEEE Trans. Med. Imaging. 34(9):1797–1807. DOI: 10.1109/TMI.2015.2409024.]
   PubMed Web of Science ®Google Scholar
• Chambolle, A., Morini, M., Ponsiglione, M. (2012). A non-local mean curvature flow and its semi-implicit time-discrete approximation. SIAM J. Math. Anal. 44(6):4048–4077. DOI: 10.1137/120863587.
   Web of Science ®Google Scholar
• Cesaroni, A., Pagliari, V. (2018). Convergence of nonlocal geometric flows to anisotropic mean curvature motion. arXiv preprint https://arxiv.org/abs/1811.01732.
   Google Scholar
• De Giorgi, E. (1994). Barriers, boundaries, motion of manifolds. Conference held at the Department of Mathematics of Pavia, March.
   Google Scholar
• Bellettini, G., Novaga, M. (1998). Comparison results between minimal barriers and viscosity solutions for geometric evolutions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26(1):97–131.
   Google Scholar
• Ishii, H., Souganidis, P. (1995). Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Tohoku Math. J. 47(2):2, 227–250. DOI: 10.2748/tmj/1178225593.
   Web of Science ®Google Scholar
• Chen, H., Weth, T. (2019). The Dirichlet problem for the logarithmic Laplacian. Comm. Partial Differ. Equations. 44(11):1100–1139. DOI: 10.1080/03605302.2019.1611851.
   Web of Science ®Google Scholar
• Valdinoci, E. (2013). A fractional framework for perimeters and phase transitions. Milan J. Math. 81(1):1–23. DOI: 10.1007/s00032-013-0199-x.
   Web of Science ®Google Scholar
• Abatangelo, N., Valdinoci, E. (2014). A notion of nonlocal curvature. Numer. Funct. Anal. Optim. 35(7–9):793–815. DOI: 10.1080/01630563.2014.901837.
   Web of Science ®Google Scholar
• Caffarelli, L., Valdinoci, E. (2013). Regularity properties of nonlocal minimal surfaces via limiting arguments. Adv. Math. 248:843–871. DOI: 10.1016/j.aim.2013.08.007.
   Web of Science ®Google Scholar
• Ludwig, M. (2014). Anisotropic fractional perimeters. J. Differ. Geom. 96(1):77–93. DOI: 10.4310/jdg/1391192693.
   Web of Science ®Google Scholar
• Dipierro, S., Novaga, M., Valdinoci, E. (2019). On a Minkowski geometric flow in the plane: evolution of curves with lack of scale invariance. J. London Math. Soc. 99(1):31–51. DOI: 10.1112/jlms.12162.
   Web of Science ®Google Scholar