Related papers: Quantitative stratification for some free-boundary…
One of the earliest quantum algorithms was discovered by Bernstein and Vazirani, for a problem called Recursive Fourier Sampling. This paper shows that the Bernstein-Vazirani algorithm is not far from optimal. The moral is that the need to…
We introduce the boundary measure at scale r of a compact subset of the n-dimensional Euclidean space. We show how it can be computed for point clouds and suggest these measures can be used for feature detection. The main contribution of…
In this work a linearly constrained minimization of a positive semidefinite quadratic functional is examined. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive operator related to the functional,…
We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free…
In this paper, we obtain \textit{quantitative} estimates on the fine structure of the singular set of the mutual boundary $\partial \Omega^{\pm}$ for pairs of complementary domains, $\Omega^+, \Omega^- \subset \mathbb{R}^n$ which arise in a…
We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T.…
In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up…
We prove the -- to the best knowledge of the authors -- first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by our recent partial…
We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear…
We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our…
We prove near-optimal trade-offs for quantifier depth versus number of variables in first-order logic by exhibiting pairs of $n$-element structures that can be distinguished by a $k$-variable first-order sentence but where every such…
We consider the following classical conjecture of Besicovitch: a $1$-dimensional Borel set in the plane with finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$ which has lower density strictly larger than $\frac{1}{2}$ almost…
We prove partial regularity of suitable weak solutions to the Navier--Stokes equations at the boundary in irregular domains. In particular, we provide a criterion which yields continuity of the velocity field in a boundary point and obtain…
We prove the multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary of dimension between 3 and 7 for generic metrics. To approach this, we develop existence and regularity theory for free…
In this paper we consider a weakly coupled $p$-Laplacian system of a Bernoulli type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free…
In this series of papers, we present a set of methods to revive quantum geometrodynamics which encountered numerous mathematical and conceptual challenges in its original form promoted by Wheeler and De Witt. In this paper, we introduce the…
We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti…
This paper studies an inverse boundary value problem for a semilinear Helmholtz equation with Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n\ge2$). The objective is to recover the unknown linear and…
Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space $\R^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers…
We study properties of positive functions satisfying (E) --$\Delta$u+m|$\nabla$u| q -- u p = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a…