Related papers: Contact points with integer frequencies in the thi…
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…
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 review the finite element approximation of the classical obstacle problem in energy and max-norms and derive error estimates for both the solution and the free boundary. On the basis of recent regularity results we present an optimal…
Diffusive search for a static target is a common problem in statistical physics with numerous applications in chemistry and biology. We look at this problem from a different perspective and investigate the statistics of encounters between…
We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from…
We consider the inverse scattering problem of retrieving the structural parameters of a stratified medium consisting of dispersive materials, given knowledge of the complex reflection coefficient in a finite frequency range. It is shown…
It is proved that if the scattering amplitudes at a fixed wavenumber for two obstacles from a certain large class of obstacles differ a little, than the obstacles differ a little. Error estimate is given. It is proved that there is an…
We consider the inverse elastic scattering problems using the far field data due to one incident plane wave. A simple method is proposed to reconstruct the location and size of the obstacle using different components of the far field…
This paper is concerned with the problem of scattering of time-harmonic acoustic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation…
The thin interface limit aims at minimizing the effects arising from a numerical interface thickness, inherent in diffuse interface models of solidification and microstructure evolution such as the phase field model. While the original…
We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches…
Convergence rates in spectral regularization methods quantify the approximation error in inverse problems as a function of the noise level or the number of sampling points. Classical strong convergence rate results typically rely on source…
This paper is concerned with the inverse obstacle scattering problem with phaseless far-field data at a fixed frequency. The main difficulty of this problem is the so-called translation invariance property of the modulus of the far-field…
This paper considers 3-D elastic scattering problems by penetrable obstacles with embedded objects. The well-posedness of transmission problem is proved by employing integral equation method. Then the Inverse Problems , which is to recover…
We are concerned with the reconstruction of a sound-soft obstacle using far field measurements of the scattered waves associated with incident plane waves sent from one direction but at multiple frequencies. We define, for each frequency,…
We give a simple proof of the fact that - in all dimensions - there are no homogeneous solutions to the thin obstacle problem with frequency $\lambda$ belonging to intervals of the form $(2k,2k+1)$, $k \in \mathbb{N}$. In particular, there…
Given two points in the plane, a set of obstacles defined by closed curves, and an integer $k$, does there exist a path between the two designated points intersecting at most $k$ of the obstacles? This is a fundamental and well-studied…
Large density fluctuations of conserved charges have been proposed as a promising signature for exploring the QCD critical point in heavy-ion collisions. These fluctuations are expected to exhibit a fractal or scale-invariant behavior,…
We study rigidity/flexibility properties of global solutions to the thin obstacle problem. For solutions with bounded positive sets, we give a classification in terms of their expansions at infinity. For solutions with bounded contact sets,…