On mean curvature flow with forcing. Issue 5 (3rd May 2020)

• Record Type: Journal Article
• Authors: Kim, Inwon; Kwon, Dohyun
• Is Part Of: Communications in partial differential equations. Volume 45:Issue 5(2020)
• Page Start: 414
• Page End: 455
• Publication Date: 2020-05-03
• Subjects: Mean curvature flow -- minimizing movements -- moving planes method -- star-shaped -- viscosity solutions; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2019.1695262
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ -reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow's exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed by Feldman-Kim to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.