# A logarithmic epiperimetric inequality for the obstacle problem

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## Abstract

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 *C*^{1} 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.

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## References

- AS88.David Adams, Leon Simon.: Rates of asymptotic convergence near isolated singularities of geometric extrema. Indiana Univ. Math. J.
**37**(2), 225–254 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - Caf98.Caffarelli L. A.: The obstacle problem revisited. J. Fourier Anal. Appl.
**4**(4-5), 383–402 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - CR76.Caffarelli L. A., Rivière N. M.: Smoothness and analyticity of free boundries in variational inequalities. Ann. Scuola Norm. Sup. Pisa Cl. Sci.
**3**(2), 289–310 (1976)MathSciNetzbMATHGoogle Scholar - Caf77.Luis A. Caffarelli: The regularity of free boundaries in higher dimensions. Acta Math.
**139**(3-4), 155–184 (1977)MathSciNetzbMATHGoogle Scholar - CSV17.M. Colombo, L. Spolaor, and B. Velichkov. Direct epiperimetric inequalities for the thin obstacle problem. arXiv:1709.03120 (2017).
- Fef09.Charles Fefferman: Extension of \(C^{m,\omega}\)-smooth functions by linear operators. Rev. Mat. Iberoam.
**25**(1), 1–48 (2009)MathSciNetGoogle Scholar - FS17.A. Figalli, J. Serra. On the fine structure of the free boundary for the classical obstacle problem.
*Preprint*, 2017.Google Scholar - FGS15.Focardi M., Gelli M. S., Spadaro E.: Monotonicity formulas for obstacle problems with Lipschitz coefficients. Calc. Var. Partial Differential Equations
**54**(2), 1547–1573 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - FGS17.Focardi M., Geraci F., Spadaro E.: The classical obstacle problem for nonlinear variational energies. Nonlinear Anal.
**154**, 71–87 (2017)MathSciNetCrossRefzbMATHGoogle 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 - GPV16.Nicola Garofalo, Arshak Petrosyan, Mariana Smit Vega Garcia: An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients. J. Math. Pures Appl.
**105**(6), 745–787 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - Ger85.Claus Gerhardt: Global
*C*^{1,1}-regularity for solutions of quasilinear variational inequalities. Arch. Rational Mech. Anal.**89**(1), 83–92 (1985)MathSciNetCrossRefzbMATHGoogle 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 - Rei64.Reifenberg E. R.: An epiperimetric inequality related to the analyticity of minimal surfaces. Ann. of Math. (2)
**80**, 1–14 (1964)MathSciNetCrossRefzbMATHGoogle 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 - Wei99a.Georg S. Weiss: A homogeneity improvement approach to the obstacle problem. Invent. Math.
**138**(1), 23–50 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - Wei99b.Georg Sebastian Weiss: Partial regularity for a minimum problem with free boundary. J. Geom. Anal.
**9**(2), 317–326 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - Whi83.Brian White: Tangent cones to two-dimensional area-minimizing integral currents are unique. Duke Math. J.
**50**(1), 143–160 (1983)MathSciNetCrossRefzbMATHGoogle Scholar