For the general obstacle problem, we prove by direct methods an epiperimetric inequality at regular and singular points, thus answering a question of Weiss (Invent. Math., 138 (1999), 23-50). In particular at singular points we introduce a new tool, which we call logarithmic epiperimetric inequality, which yields an explicit logarithmic modulus of continuity on the C 1 regularity of the singular set, thus improving previous results of Caffarelli and Monneau [4,2,11].

We introduce a new logarithmic epiperimetric inequality for the 2m‐Weiss energy in any dimension, and we recover with a simple direct approach the usual epiperimetric inequality for the 3/2‐Weiss energy. In particular, even in the latter case, unlike the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension 2, we also prove for the first time the classical epiperimetric inequality for the (2m − 1/2)‐Weiss energy, thus covering all the admissible energies. As a first application, we classify the global λ‐homogeneous minimizers of the thin obstacle problem, with λ∈[]3/2,2+c∪⋃m∈ℕ‍(),2m−cm−2m+cm+, showing as a consequence that the frequencies 3/2 and 2m are isolated and thus improving on the previously known results. Moreover, we give an example of a new family of (2m − 1/2)‐homogeneous minimizers in dimension higher than 2. Second, we give a short and self‐contained proof of the regularity of the free boundary of the thin obstacle problem, previously obtained by Athanasopoulos, Caffarelli, and Salsa (2008) for regular points and Garofalo and Petrosyan (2009) for singular points. In particular, we improve the C1 regularity of the singular set with frequency 2m by an explicit logarithmic modulus of continuity. © 2019 Wiley Periodicals, Inc.

Using a direct approach, we prove a two‐dimensional epiperimetric inequality for the one‐phase problem in the scalar and vectorial cases and for the double‐phase problem. From this we deduce, in dimension 2, the C1,α regularity of the free boundary in the scalar one‐phase and double‐phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of {|u|=0} might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc.

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.

We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Lojasiewicz inequality. The difficulty lies in the fact that, since the constraint is nonanalytic, the pioneering method of L. Simon ([25]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.In the seminal paper [25], Leon Simon proved (1.3) for the flow associated to harmonic maps between two analytic manifolds. Notice that, also in this case, there is a geometric constraint given by the target manifold, but a change of coordinates allows to trivialize this constraint, while transforming the Dirichlet energy into an analytic functional F. In [25], Simon showed that the