Related papers: Microlocal analysis of singular measures
We are concerned with the Calder\'on inverse inclusion problem, where one intends to recover the shape of an inhomogeneous conductive inclusion embedded in a homogeneous conductivity by the associated boundary measurements. We consider the…
Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a…
In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The…
For a non-elementary subgroup of the mapping class group of a surface, we study its invariant Radon measures on the space of measured laminations, by classifying them on the recurrent measured laminations. In particular, given a…
We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and…
We extend both the Hawking-Penrose Theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity $C^1$. For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the…
Our aim is to explain instances in which the value of the logarithmic Mahler measure of a polynomial can be written in an unexpectedly neat manner. To this end we examine polynomials defining rational curves, which allows their zero-locus…
The art of analysis involves the subtle combination of approximation, inequalities, and geometric intuition as well as being able to work at different scales. With this subtlety in mind, we present this paper in a manner designed for wide…
In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2\pi)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the K\"ahler-Einstein equations actually possess maximum…
For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly-decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions:…
For ergodic 1d Jacobi operators we prove that the random singular components of any spectral measure are almost surely mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's…
Following theoretical and experimental work of Maas et al we consider a linearized model for internal waves in effectively two dimensional aquaria. We provide a precise description of singular profiles appearing in long time wave evolution…
We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is…
We investigate elliptic irregular obstacle problems with $p$-growth involving measure data. Emphasis is on the strongly singular case $1 < p \le 2-1/n$, and we obtain several new comparison estimates to prove gradient potential estimates in…
This paper is concerned with locally damped semilinear wave equations defined on compact Riemannian manifolds with boundary. We present a construction of measure-controlled damping regions which are sharp in the sense that their summed…
The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is know as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of…
The geometry of impulsive pp-waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric. The geodesic equation involves formally ill-defined products of distributions due…
We show $\ell^p\big(\mathbb Z^d\big)$ boundedness, for $p\in(1, \infty)$, of discrete singular integrals of Radon type with the aid of appropriate square function estimates, which can be thought as a discrete counterpart of the…
We demonstrate that morphological observables (e.g. steepness of the radial light profile, ellipticity, asymmetry) are intertwined and cannot be measured independently of each other. We present strong arguments in favour of model-based…
This paper studies complete non-compact smooth metric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with positive first spectrum $\lambda_1(\Delta_f)$ or satisfying a weighted Poincar\'e inequality with weight function $\rho$. We…