Abstract
Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1/(n+1)-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1/2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparison result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.
Highlights
Many processes in material sciences such as phase transformation, crystal growth, grain growth, stress-driven rearrangement instabilities, etc., can be modelled as geometric interface motions, in which surface tensions act as a principal driving force
Where V denotes the normal velocity of Γt in the direction of the unit outer normal ν of Γt and β is the mobility, a positive kinetic coefficient [29]
The aim of the present paper is to prove the existence of a generalized minimizing movement (GMM) for anisotropic mean curvature flow of partitions with no restrictions on the space dimension, in the presence of a set of mobilities and forcing terms, and to point out some qualitative properties of this weak evolution, which are obtained via a comparison argument with a GMM of each single phase considered separately
Summary
Many processes in material sciences such as phase transformation, crystal growth, grain growth, stress-driven rearrangement instabilities, etc., can be modelled as geometric interface motions, in which surface tensions act as a principal driving force (see e.g., [15, 40, 49, 51] and references therein). The aim of the present paper is to prove the existence of a GMM for anisotropic mean curvature flow of partitions with no restrictions on the space dimension, in the presence of a set of mobilities and forcing terms, and to point out some qualitative properties of this weak evolution, which are obtained via a comparison argument with a GMM of each single phase considered separately. Let us mention that a natural problem remains open, namely the consistency of GMM with the classical solution, provided the latter exists, at least on a short time interval Such a result has been proven by Almgren–Taylor–Wang in [1] in the two-phase case without mobility; the proof is based on various stability properties of the flow, and using comparison arguments.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
