Related papers: A note on the frequency gaps between integers in t…

For all odd positive integers $m$, we construct $\mu$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^3,$ with $\mu\in(m,m+1)$. For $m$ large, $\mu-m$ converges to $1$, so $\mu\neq m+\tfrac 1 2$. The restriction to odd…

In this paper, we consider the thin obstacle problem in $\mathbb{R}^2$ with data at infinity. We first prove the existence and uniqueness of it. Then we show that its symmetric solutions are actually half-space solutions. Our results are…

The key point to prove the optimal $C^{1,\frac12}$ regularity of the thin obstacle problem is that the frequency at a point of the free boundary $x_0\in\Gamma(u)$, say $N^{x_0}(0^+,u)$, satisfies the lower bound $N^{x_0}(0^+,u)\ge\frac32$.…

For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency $\frac32$ and $2m$, for $m\in \mathbb N$, satisfy an epiperimetric inequality, in the latter case of logarithmic type.…

In this paper we show that, if an increasing sequence $\Lambda=(\lambda_k)_{k\in\mathbb{Z}}$ has gaps going to infinity $\lambda_{k+1}-\lambda_k\to +\infty$ when $k\to\pm\infty$, then for every $T>0$ and every sequence…

For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence…

We prove for the first time an epiperimetric inequality for the thin obstacle Weiss' energy with odd frequencies and we apply it to solutions to the thin obstacle problem with general $C^{k,\gamma}$. In particular, we obtain the rate of…

Examples are constructed of sparse subsequences of the integers for which the associated maximal averages operator is of weak type (1,1). A consequence, by transference, is that an almost everywhere L^1 -- type ergodic theorem holds for…

We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to…

The three gap theorem was originally a conjecture by Steinhaus, who asserted that there are at most three distinct gap lengths in the fractional parts of the sequence {\alpha},{2}{\alpha},{\cdots},{N}{\alpha} for any integer {N} and real…

We study uncertainty principles for orthonormal bases and sequences in $L^2(\R^d)$. As in the classical Heisenberg inequality we focus on the product of the dispersions of a function and its Fourier transform. In particular we prove that…

The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane $\Gamma$ in $\mathbb{R}^n$ and a periodic perforation $\mathcal{T}_\varepsilon$ of $\mathbb{R}^n$, depending on a…

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in…

We consider the problem of constrained Ginibre ensemble with prescribed portion of eigenvalues on a given curve $\Gamma\subset \mathbb R^2$ and relate it to a thin obstacle problem. The key step in the proof is the $H^1$ estimate for the…

This article provides optimal constants for two quantitative recurrence problems. First of all for recurrence of maps of the interval [0,1] that preserve the Lebesgue measure and on the other hand Lagrange spectrum of interval exchange…

In this paper, we show that for any integer $a \geq 2$, each of the intervals $[a^k , a^{k + 1})$ ($k \in \mathbb{N}$) contains either $\left\lfloor \frac{\log a}{\log\Phi}\right\rfloor$ or $\left\lceil \frac{\log a}{\log\Phi}\right\rceil$…

For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at all free boundary points, with the exception of a $(n-3)$-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion…

We find bounds for the maximal length of a sequence of distinct $\bf{\Gamma_{2n+1,m}}$-sets under $AD$ and show there is no sequence of distinct $\bf{\Gamma_{2n+1}}$-sets of length $\bf{\delta^1_{2n+3}}$. As a special case, there is no…

The thin obstacle problem or $n$-dimensional Signorini problem is a classical variational problem arising in several applications, starting with its first introduction in elasticity theory. The vast literature concerns mostly quadratic…

Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong…