Existence and regularity theorems of one-dimensional Brakke flows
Abstract
Given a closed countably $1$-rectifiable set in $\mathbb R^2$ with locally finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class $W^{2,2}$ whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.09763
- arXiv:
- arXiv:2004.09763
- Bibcode:
- 2020arXiv200409763K
- Keywords:
-
- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- 53C44;
- 49Q20;
- 49N60
- E-Print:
- 47 pages, 4 figures, to appear from Interfaces and Free Boundaries