arXiv subjects

Franck Wielonsky

Publications and source records attributed to Franck Wielonsky.

10 recordsLinked to original sources

arXiv2025

Riesz equilibrium on a ball in the external field of a point charge

We investigate the Riesz energy minimization problem on a $d$-dimensional ball in the presence of an external field created by a point charge above the ball in $\R^{d+1}$, $d\geq1$. Both cases of an attractive charge and a repulsive charge are considered. The notion of a signed equilibrium measure is one of the main tools in the present study. For the case of a positive (repulsive) charge, the determination of the support of the equilibrium measure is a nontrivial question. We solve it in the one-dimensional case by making use of iterated balayage, a method already applied in logarithmic potential theory. Here we use a modified version of it, in order to handle the phenomenon of mass loss, characteristic of the Riesz balayage of positive measures. Moreover, we also consider minimization of Coulomb energy on the ball in dimension $d\geq2$, and of logarithmic energy on the segment in dimension 1. Different techniques are used for these two cases.

math.CA↗

arXiv2022

Riesz energy problems with external fields and related theory

In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ assumed in several previous studies. We provide sufficient conditions on $Q$ for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor $\R^{d}$, embedded in $\R^{d+1}$, when the external field is created by the potential of a signed measure $ν$ outside of $\R^{d}$. Simple cases where $ν$ is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vallée-Poussin, are established. These results are of independent interest.

math.CA↗

arXiv2019

Equilibrium problems in weakly admissible external fields created by pointwise charges

The main subject of this paper is equilibrium problems on an unbounded conductor $Σ$ of the complex plane in the presence of a weakly admissible external field. An admissible external field $Q$ on $Σ$ satisfies, along with other mild conditions, the following growth property at infinity: $$\lim_{|x| \rightarrow \infty}(Q(x) - \log |x|) = +\infty.$$ This condition guarantees the existence and uniqueness of the equilibrium measure in the presence of $Q$, and the compactness of its support. In the last 10-15 years, several papers have dealt with weakly admissible external fields, in the sense that $Q$ satisfies a weaker condition at infinity, namely, $$\exists M\in(-\infty,\infty],\quad\liminf_{|x| \rightarrow \infty}(Q(x) - \log |x|) = M.$$ Under this last assumption, there still exists a unique equilibrium measure in the external field $Q$, but the support need not be a compact subset of $Σ$ anymore. In most examples considered in the literature the support is indeed unbounded. Our main goal in this paper is to illustrate this topic by means of a simple class of external fields on the real axis created by a pair of attractive and repellent charges in the complex plane, and to study the dynamics of the associated equilibrium measures as the strength of the charges evolves. As one of our findings, we exhibit configurations where the support of the equilibrium measure in a weakly admissible external field is a compact subset of the real axis. To achieve our goal, we extend some results from potential theory, known for admissible external fields, to the weakly admissible case. These new results may be of independent interest. Finally, the so--called signed equilibrium measure is an important tool in our analysis. Its relationship with the (positive) equilibrium measure is also explored.

math.CV↗

arXiv2016

A large deviation principle for weighted Riesz interactions

We prove a large deviation principle for the sequence of push-forwards of empirical measures in the setting of Riesz potential interactions on compact subsets K in R^d with continuous external fields. Our results are valid for base measures on K satisfying a strong Bernstein-Markov type property for Riesz potentials. Furthermore, we give sufficient conditions on K (which are satisfied if K is a smooth submanifold) so that a measure on K which satisfies a mass-density condition will also satisfy this strong Bernstein-Markov property.

math.CA↗

arXiv2015

Bernstein-Markov: a survey

We give a survey of recent results, due mainly to the authors, concerning Bernstein-Markov type inequalities and connections with potential theory.

math.CV↗

arXiv2015

A boundary Problem for conjugate conductivity equations

We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half-plane with conductivity $σ(x,y)=x^p$, $p\in\mathbb{Z}$. The representations are obtained via a Riemann-Hilbert problem on the complex plane when $p$ is even and on a two-sheeted Riemann surface when $p$ is odd. They involve the Dirichlet and Neumann data on the boundary of the domain. We also show how to make the conversion from one type of conditions to the other by using the so-called global relation. The method used to derive our integral representations could be applied in any bounded simply-connected domain of the right half-plane with a smooth boundary.

math.CV↗

arXiv2014

Generic properties of Padé approximants and Padé universal series

We establish properties concerning the distribution of poles of Pad e approximants, which are generic in Baire category sense. We also investigate Pad e universal series, an analog of classical universal series, where Taylor partial sums are replaced with Pad e approximants. In particular, we complement previous studies on this subject by exhibiting dense or closed infi nite dimensional linear subspaces of analytic functions in a simply connected domain of the complex plane, containing the origin, whose all non zero elements are made of Pad e universal series. We also show how Pad e universal series can be built from classical universal series with large Ostrowski-gaps.

math.FA↗

arXiv2012

Optimal logarithmic estimates in the Hardy-Sobolev space of the disk and stability results

We prove a logarithmic estimate in the Hardy-Sobolev space $H^{k, 2}$, $k$ a positive integer, of the unit disk ${\mathbb D}$. This estimate extends those previously established by L. Baratchart and M. Zerner in $H^{1,2}$ and by S. Chaabane and I. Feki in $H^{k,\infty}$. We use it to derive logarithmic stability results for the inverse problem of identifying Robin's coefficients in corrosion detection by electrostatic boundary measurements and for a recovery interpolation scheme in the Hardy-Sobolev space $H^{k, 2}$ with interpolation points located on the boundary ${\mathbb T}$ of the unit disk.

math.CA↗

arXiv2012

Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses

We prove existence and uniqueness of a solution to the problem of minimizing the logarithmic energy of vector potentials associated to a $d$-tuple of positive measures supported on closed subsets of the complex plane. The assumptions we make on the interaction matrix are weaker than the usual ones and we also let the masses of the measures vary in a compact subset of $\R_+^d$. The solution is characterized in terms of variational inequalities. Finally, we review a few examples taken from the recent literature that are related to our results.

math.CA↗

arXiv2005

Optimal superluminal systems

We demonstrate that significant effects in the "superluminal propagation"of light-pulses cannot be observed without involving systems whose gain explodes outside the pulse spectrum. We explicitly determine the minimum norm of the gain to attain given superluminal effects and the transfer function of the corresponding optimal system. The gain-norms which would be required with the most efficient systems considered up to now (dispersive media, photonic barriers) to attain the same effects are shown to exceed the minimum by several orders of magnitude. We finally estimate the largest superluminal advances which could be attained in a realistic experiment.

physics.optics↗