A logarithmic epiperimetric inequality for the obstacle problem
- 101 Downloads
We study the regularity of the regular and of the singular set of the obstacle problem in any dimension. Our approach is related to the epiperimetric inequality of Weiss (Invent Math 138:23–50, Wei99a), which works at regular points and provides an alternative to the methods previously introduced by Caffarelli (Acta Math 139:155–184, Caf77). In his paper, Weiss uses a contradiction argument for the regular set and he asks the question if such epiperimetric inequality can be proved in a direct way (namely, exhibiting explicit competitors), which would have significant implications on the regularity of the free boundary in dimension d > 2. We answer positively the question of Weiss, proving at regular points the epiperimetric inequality in a direct way, and more significantly we introduce a new tool, which we call logarithmic epiperimetric inequality. It allows to study the regularity of the whole singular set and yields an explicit logarithmic modulus of continuity on the C1 regularity, thus improving previous results of Caffarelli and Monneau and providing a fully alternative method. It is the first instance in the literature (even in the context of minimal surfaces) of an epiperimetric inequality of logarithmic type and the first instance in which the epiperimetric inequality for singular points has a direct proof. Our logarithmic epiperimetric inequality at singular points has a quite general nature and will be applied to provide similar results in different contexts, for instance for the thin obstacle problem.
Unable to display preview. Download preview PDF.
- CSV17.M. Colombo, L. Spolaor, and B. Velichkov. Direct epiperimetric inequalities for the thin obstacle problem. arXiv:1709.03120 (2017).
- FS17.A. Figalli, J. Serra. On the fine structure of the free boundary for the classical obstacle problem. Preprint, 2017.Google Scholar
- FS16.Matteo Focardi, Emanuele Spadaro. An epiperimetric inequality for the thin obstacle problem. Adv. Differential Equations, 21(1-2):153–200, 2016.Google Scholar
- Mon01.Regis Monneau. A brief overview on the obstacle problem. In European Congress of Mathematics, Vol. II (Barcelona, 2000), volume 202 of Progr. Math., pages 303–312. Birkhäuser, Basel, 2001.Google Scholar
- SV16.L. Spolaor and B. Velichkov. An epiperimetric inequality for the regularity of some free boundary problems: the 2-dimensional case. Comm. Pure Appl. Math. (to appear)Google Scholar