Logarithmic Capacity Research Articles

Number of components of polynomial lemniscates: A problem of Erdös, Herzog, and Piranian

Let K⊂C be a compact set in the plane whose logarithmic capacity c(K) is strictly positive. Let Pn(K) be the space of monic polynomials of degree n, all of whose zeros lie in K. For p∈Pn(K), its filled unit leminscate is defined by Λp={z:|p(z)|<1}. Let C(Λp) denote the number of connected components of the open set Λp, and define Cn(K)=maxp∈Pn(K)⁡C(Λp). In this paper we show that the quantityM(K)=limsupn→∞Cn(K)n, satisfies M(K)<1 when the logarithmic capacity c(K)<1, and M(K)=1 when c(K)≥1. In particular, this answers a question of Erdös et al. posed in (1958) [8]. In addition, we show that for nice enough compact sets whose capacity is strictly bigger than 12, the quantity m(K)=liminfn→∞Cn(K)n>0.

Capacities, Green function, and Bergman functions

AbstractUsing the logarithmic capacity, we give quantitative estimates of the Green function, as well as precise lower bounds of the Bergman kernel for bounded pseudoconvex domains in and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies the sharp estimate if is a bounded pseudoconvex domain with ‐boundary.

Logarithmic capacity under holomorphic motions in higher dimensions

We show that logarithmic capacity in even dimensional Euclidean spaces varies subharmonically under holomorphic motions.

Computing the logarithmic capacity of compact sets having (infinitely) many components with the charge simulation method

We apply the charge simulation method (CSM) in order to compute the logarithmic capacity of compact sets consisting of (infinitely) many “small” components. This application allows to use just a single charge point for each component. The resulting method therefore is significantly more efficient than methods based on discretizations of the boundaries (for example, our own method presented in Liesen et al. (Comput. Methods Funct. Theory17, 689–713, 2017)), while maintaining a very high level of accuracy. We study properties of the linear algebraic systems that arise in the CSM, and show how these systems can be solved efficiently using preconditioned iterative methods, where the matrix-vector products are computed using the fast multipole method. We illustrate the use of the method on generalized Cantor sets and the Cantor dust.

On the Hilbert problem for semi-linear Beltrami equations

The presented paper is devoted to the study of the well-known Hilbert boundary-value problem for semi-linear Beltrami equations with arbitrary boundary data that are measurable with respect to logarithmic capacity. Namely, we prove here the corresponding results on the existence, regularity, and representation of its nonclassical solutions with a geometric interpretation of boundary values as the angular (along the nontangential paths) limits in comparison with the classical approach in PDE. For this purpose, we apply completely continuous operators by Ahlfors-Bers, first of all to obtain solutions of semi-linear Beltrami equations, generally speaking with no boundary conditions, and then to derive their representation through the solutions of the Vekua-type equations and the so-called generalized analytic functions with sources. Besides, we obtain similar results for nonclassical solutions of the Poincare boundary-value problem on directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to semi-linear equations of the Poisson type. The obtained results are applied to some problems of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states, and stationary burning in anisotropic and inhomogeneous media.

The Lebesgue constants for Leja points are subexponential

It is shown that on a set of positive logarithmic capacity the Lebesgue constants associated with Leja points are subexponential. Actually, the same result is verified for the so-called τ-Leja points introduced by Andrievskii and Nazarov.

Rigidity Theorems by Capacities and Kernels

Abstract For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the logarithmic capacity $c_{\beta }$, and the analytic capacity $c_{B}$ satisfy the inequality chain $\pi K \geq c^2_{\beta } \geq c^2_B$. Moreover, equality holds at a single point between any two of the three quantities if and only if $X$ is biholomorphic to a disk possibly less a relatively closed polar set. We extend the inequality chain by showing that $c_{B}^2 \geq \pi v^{-1}(X)$ on planar domains, where $v(\cdot )$ is the Euclidean volume, and characterize the extremal cases when equality holds at one point. Similar rigidity theorems concerning the Szegö kernel, the higher-order Bergman kernels, and the sublevel sets of the Green’s function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture.

Poincaré problem with measurable data for semilinear Poisson equation in the plane

We study the Poincaré boundary-value problem with measurable in terms of the logarithmic capacity boundary data for semilinear Poisson equations defined either in the unit disk or in Jordan domains with quasihyperbolic boundary condition. The solvability theorems as well as their applications to some semilinear equations, modelling diffusion with absorption, plasma states and stationary burning, are given.

Walsh’s Conformal Map onto Lemniscatic Domains for Polynomial Pre-images I

We consider Walsh’s conformal map from the exterior of a compact set E subseteq mathbb {C} onto a lemniscatic domain. If E is simply connected, the lemniscatic domain is the exterior of a circle, while if E has several components, the lemniscatic domain is the exterior of a generalized lemniscate and is determined by the logarithmic capacity of E and by the exponents and centers of the generalized lemniscate. For general E, we characterize the exponents in terms of the Green’s function of E^c. Under additional symmetry conditions on E, we also locate the centers of the lemniscatic domain. For polynomial pre-images E = P^{-1}(Omega ) of a simply-connected infinite compact set Omega , we explicitly determine the exponents in the lemniscatic domain and derive a set of equations to determine the centers of the lemniscatic domain. Finally, we present several examples where we explicitly obtain the exponents and centers of the lemniscatic domain, as well as the conformal map.

On some variational problems involving capacity, torsional rigidity, perimeter and measure

Abstract We investigate the existence of a maximiser among open, bounded, convex sets in R d \mathbb{R}^{d} , d ≥ 3 d\geq 3 , for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if d = 2 d=2 ), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter.

Riesz capacities of a set due to Dobiński

We study the Riesz $(a,p)$-capacity of the so called Dobi\'nski set. We characterize the values of the parameters $a$ and $p$ for which the $(a,p)$-Riesz capacity of the Dobi\'nski set is positive. In particular we show that the Dobi\'nski set has positive logarithmic capacity, thus answering a question of Dayan, Fernand\'ez and Gonz\'alez. We approach the problem by considering the dyadic analogues of the Riesz $(a,p)$-capacities which seem to be better adapted to the problem.

Automation of reversible steganographic coding with nonlinear discrete optimisation

Authentication mechanisms are at the forefront of defending the world from various types of cybercrime. Steganography can serve as an authentication solution through the use of a digital signature embedded in a carrier object to ensure the integrity of the object and simultaneously lighten the burden of metadata management. Nevertheless, despite being generally imperceptible to human sensory systems, any degree of steganographic distortion might be inadmissible in fidelity-sensitive situations such as forensic science, legal proceedings, medical diagnosis and military reconnaissance. This has led to the development of reversible steganography. A fundamental element of reversible steganography is predictive analytics, for which powerful neural network models have been effectively deployed. Another core element is reversible steganographic coding. Contemporary coding is based primarily on heuristics, which offers a shortcut towards sufficient, but not necessarily optimal, capacity–distortion performance. While attempts have been made to realise automatic coding with neural networks, perfect reversibility is unattainable via such learning machinery. Instead of relying on heuristics and machine learning, we aim to derive optimal coding by means of mathematical optimisation. In this study, we formulate reversible steganographic coding as a nonlinear discrete optimisation problem with a logarithmic capacity constraint and a quadratic distortion objective. Linearisation techniques are developed to enable iterative mixed-integer linear programming. Experimental results validate the near-optimality of the proposed optimisation algorithm when benchmarked against a brute-force method.

Hilbert problem with measurable data for semilinear equations of the Vekua type

We prove the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the nonlinear equations of the Vekua’s type ∂z f (z ) = h (z )q(f (z )). The found solutions differ from the classical ones, because our approach is based on the notion of boundary values in the sense of angular limits along nontangential paths. The results obtained can be applied to the establishment of existence theorems for the Poincaré and Neumann boundary-value problems for the nonlinear Poisson equations of the form ΔU (z )=H (z )Q(U (z )) with arbitrary measurable boundary data with respect to the logarithmic capacity. They can be also applied to the study of some semilinear equations of mathematical physics modeling such processes as the diffusion with absorption, plasma states, stationary burning etc. in anisotropic and inhomogeneous media.

Totally null sets and capacity in Dirichlet type spaces

In the context of Dirichlet type spaces on the unit ball of C d $\mathbb {C}^d$ , also known as Hardy–Sobolev or Besov–Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthening of boundary interpolation theorems of Peller and Khrushchëv and of Cohn and Verbitsky.

Dirichlet problem with measurable data for semilinear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk  goes to the known dissertation of Luzin. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. With a view to further developments of the theory of boundary-value problems for semilinear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is firstly constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations ΔU =G over the sources G ∈Lp, p &gt;1. The latter makes it possible to apply the Leray— Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form ΔU (z ) =H (z )⋅Q(U (z )) for multipliers H ∈Lp with p &gt;1 and continuous functions Q: R→R with Q(t ) /t →0 as t →∞. These results can be applied to some specific quasilinear equations of mathematical physics, arising under a modeling of various physical processes such as the diffusion with absorption, plasma states, stationary burning, etc. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

On the Orlicz Minkowski problem for logarithmic capacity

In this paper, studying the Orlicz Minkowski problem for logarithmic capacity, we prove the existence of its solution for general measure by the method of approximation. To achieve the purpose, we first prove the existence of its solution for the special case of discrete measure by variational method.

Minimizing capacity among linear images of rotationally invariant conductors

Logarithmic capacity is shown to be minimal for a planar set having N-fold rotational symmetry (\(N \ge 3\)), among all conductors obtained from the set by area-preserving linear transformations. Newtonian and Riesz capacities obey a similar property in all dimensions, when suitably normalized linear transformations are applied to a set having irreducible symmetry group. A corollary is Pólya and Schiffer’s lower bound on capacity in terms of moment of inertia.

Dirichlet problem with measurable data for quasilinear Poisson equations

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb D$ goes to the famous dissertation of Luzin, see e.g. its reprint \cite{L}. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. Following this way, we proved in \cite{GNR1} Theorem 7 on the solvability of the Dirichlet problem for the Poisson equations $\triangle\, U=G$ with sources in classes $G\in L^p,$ $ p&gt;1$, in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. There we assumed that the domains satisfy the quasihyperbolic boundary condition by Gehring--Martio, generally speaking, without the known $(A)-$condition by Ladyzhenskaya--Ural'tseva and, in particular, without the outer cone condition that were standard for boundary-value problems in the PDE theory. Note that such Jordan domains cannot be even locally rectifiable. With a view to further development of the theory of boundary value problems for semi-linear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is first constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations $\triangle\, U=G$ with the sources $G\in L^p,$ $ p&gt;1$. The latter makes it possible to apply the Leray-Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form $\triangle\, U(z)=H(z)\cdot Q(U(z))$ for multipliers $H\in L^p$ with $ p&gt;1$ and continuous functions $Q: \mathbb R\to\mathbb R$ with $Q(t)/t\to 0$ as $t\to \infty$. As consequences, we give applications to some concrete quasilinear equations of mathematical physics, arising under modelling various phy\-si\-cal processes such as diffusion with absorption, plasma states, stationary burning etc. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Log-lightning computation of capacity and Green's function

See Video Abstract (click the "Video Abstract" button next to the "PDF" button) A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces.