Logarithmic Capacity Research Articles

To the theory of semilinear equations in the plane

In two dimensions, we present a new approach to the study of the semilinear equations of the form \(\mathrm{div}[ A(z) \nabla u] = f(u)\), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions \(A(z)\), whereas its reaction term \(f(u)\) is a continuous non-linear function. Assuming that \(f(t)/t\to 0\) as \(t\to\infty\), we establish a theorem on existence of weak \(C(\overline D)\cap W^{1,2}_{\rm loc}(D)\) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains \(D\) without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.

Hereditarily non uniformly perfect sets

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an example of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.

On Hölder continuity of solutions of the Beltrami equations on the boundary

In the present paper, it is found conditions on the complex coefficient of the Beltrami equations with the degeneration of the uniform ellipticity in the unit disk under which their generalized homeomorphic solutions are continuous by Hölder on the boundary. These results can be applied to the investigations of various boundary value problems for the Beltrami equations. In a series of recent papers, under the study of the boundary value problems of Dirichlet, Hilbert, Neumann, Poincare and Riemann with arbitrary measurable boundary data for the Beltrami equations as well as for the generalizations of the Laplace equation in anisotropic and inhomogeneous media, it was applied the logarithmic capacity, see e.g. Gutlyanskii V., Ryazanov V., Yefimushkin A. On the boundary value problems for quasiconformal functions in the plane // Ukr. Mat. Visn. - 2015. - 12, no. 3. - P. 363-389; transl. in J. Math. Sci. (N.Y.) - 2016. - 214, no. 2. - P. 200-219; Gutlyanskii V., Ryazanov V., Yefimushkin A. On a new approach to the study of plane boundary-value problems // Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. - 2017. - No. 4. - P. 12-18; Yefimushkin A. On Neumann and Poincare Problems in A-harmonic Analysis // Advances in Analysis. - 2016. - 1, no. 2. - P. 114-120; Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations in quasidisks // Ukr. Mat. Visn. - 2015. - 12, no. 2. - P. 190–209; transl. in J. Math. Sci. (N.Y.) - 2015. - 211, no. 5. - P. 646–659; Yefimushkin A., Ryazanov V. On the Riemann–Hilbert Problem for the Beltrami Equations // Contemp. Math. - 2016. - 667. - P. 299-316; Gutlyanskii V., Ryazanov V., Yakubov E., Yefimushkin A. On Hilbert problem for Beltrami equation in quasihyperbolic domains // ArXiv.org: 1807.09578v3 [math.CV] 1 Nov 2018, 28 pp. As well known, the logarithmic capacity of a set coincides with the so-called transfinite diameter of the set. This geometric characteristic implies that sets of logarithmic capacity zero and, as a consequence, measurable functions with respect to logarithmic capacity are invariant under mappings that are continuous by Hölder. That circumstance is a motivation of our research. Let \(D\) be a domain in the complex plane \(\mathbb C\) and let \(\mu: D\to\mathbb C\) be a measurable function with \( |\mu(z)| \lt 1 \) a.e. The equation of the form \(f_{\bar{z}}\ =\ \mu(z) f_z \) where \( f_{\bar z}={\bar\partial}f=(f_x+if_y)/2 \), \(f_{z}=\partial f=(f_x-if_y)/2\), \(z=x+iy\), \( f_x \) and \( f_y \) are partial derivatives of the function \(f\) in \(x\) and \(y\), respectively, is said to be a Beltrami equation. The function \(\mu\) is called its complex coefficient, and \( K_{\mu}(z)=\frac{1+|\mu(z)|}{1-|\mu(z)|}\) is called its dilatation quotient. The Beltrami equation is said to be degenerate if \({\rm ess}\,{\rm sup}\,K_{\mu}(z)=\infty\). The existence of homeomorphic solutions in the Sobolev class \(W^{1,1}_{\rm loc}\) has been recently established for many degenerate Beltrami equations under the corresponding conditions on the dilatation quotient \(K_{\mu}\), see e.g. the monograph Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami equation. A geometric approach. Developments in Mathematics, 26. Springer, New York, 2012 and the further references therein. The main theorem of the paper, Theorem 1, states that a homeomorphic solution \( f:\mathbb D\to\mathbb D \) in the Sobolev class \( W^{1,1}_{\rm loc} \) of the Beltrami equation in the unit disk \(\mathbb D\) has a homeomorphic extension to the boundary that is Hölder continuous if \(K_{\mu}\in L^1(\Bbb D)\) and, for some \(\varepsilon_0\in(0,1)\) and \(C\in[1,\infty)\), $$ \sup\limits_{\varepsilon\in(0,\varepsilon_0)} \int_{\mathbb D\cap D(\zeta,\varepsilon)}K_{\mu}(z) dm(z) \lt C \qquad \forall \zeta \in \partial \mathbb{D} $$ where \(D(\zeta,\varepsilon)=\left\{z\in{\Bbb C}: |z-\zeta| \lt \varepsilon\right\}\).

Dirichlet problem for Poisson equations in Jordan domains

First, we study the Dirichlet problem for the Poisson equations \(\triangle u(z) = g(z)\) with \(g\in L^p\), \(p>1\), and continuous boundary data \(\varphi :\partial D\to\mathbb{R}\) in arbitrary Jordan domains \(D\) in \(\mathbb{C}\) and prove the existence of continuous solutions \(u\) of the problem in the class \(W^{2,p}_{\rm loc}\). Moreover, \(u\in W^{1,q}_{\rm loc}\) for some \(q>2\) and \(u\) is locally Hölder continuous. Furthermore, \(u\in C^{1,\alpha}_{\rm loc}\) with \(\alpha = (p-2)/p\) if \(p>2\). Then, on this basis and applying the Leray-Schauder approach, we obtain the similar results for the Dirichlet problem with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form \(\triangle u(z) = h(z)\cdot f(u(z))\) with the same assumptions on \(h\) as for \(g\) above and continuous functions \(f:\mathbb{R}\to\mathbb{R}\), either bounded or with nondecreasing \(|f\,|\) of \( |t\,|\) such that \(f(t)/t \to 0\) as \(t\to\infty\). We also give here applications to mathematical physics that are relevant to problems of diffusion with absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson equations in the unit disk \(\mathbb{D}\subset\mathbb{C}\) with arbitrary boundary data \(\varphi :\partial\mathbb{D}\to\mathbb{R}\) that are measurable with respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions \(u\) of the problem in terms of the angular limits in \(\mathbb D\) a.e. on \(\partial\mathbb D\) with respect to logarithmic capacity with the same local properties as above. Finally, we extend these results to almost smooth Jordan domains with qusihyperbolic boundary condition by Gehring-Martio.

Regularity for generators of invariant subspaces of the Dirichlet shift

Let D denote the classical Dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable. For a set E⊆∂D we writeDE={f∈D:limr→1−⁡f(reit)=0q.e.}, where q.e. stands for “except possibly for eit in a set of logarithmic capacity 0”. We show that if E is a Carleson set, then there is a function f∈DE that is also in the disc algebra and that generates DE in the sense that DE=clos{pf:p is a polynomial}.We also show that if φ∈D is an extremal function (i.e. 〈pφ,φ〉=p(0) for every polynomial p), then the limits of |φ(z)| exist for every eit∈∂D as z approaches eit from within any polynomially tangential approach region.

Boundary integral equations for the exterior Robin problem in two dimensions

We propose two methods based on boundary integral equations for the numerical solution of the planar exterior Robin boundary value problem for the Laplacian in a multiply connected domain. The methods do not require any a-priori information on the logarithmic capacity. Investigating the properties of the integral operators and employing the Riesz theory we prove that the obtained boundary integral equations for both methods are uniquely solvable. The feasibility of the numerical methods is illustrated by examples obtained via solving the integral equations by the Nyström method based on weighted trigonometric quadratures on an equidistant mesh.

On the honeycomb conjecture for a class of minimal convex partitions

We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every n ∈ N n \in \mathbb {N} the minimizers of F F among convex n n -gons with given area. In particular, our result allows us to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover, we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of λ 1 \lambda _1 among convex pentagons, hexagons, and heptagons with prescribed area.

One dimensional estimates for the Bergman kernel and logarithmic capacity

Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non simply connected domains, generalizing a recent example of Forn\ae ss.

Symmetric contours and convergent interpolation

The essence of Stahl–Gonchar–Rakhmanov theory of symmetric contours as applied to the multipoint Padé approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is “symmetric” with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single-valued continuation with non-identically zero jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall and Singh, 1977; Baratchart and Yattselev, 2009). We consider rational interpolants with free poles of Cauchy transforms of non-vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize ∂̄-extension of the Riemann–Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation.

New criteria for uniform approximability by harmonic functions on compact sets in ℝ2

New uniform approximability criteria formulated in terms of logarithmic capacity are obtained for approximations by harmonic functions on compact sets in ℝ2. A relationship between these approximations and analogous approximations on compact sets in ℝ3 is established.

Fast and Accurate Computation of the Logarithmic Capacity of Compact Sets

We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it.

Eigenvalue clustering, control energy, and logarithmic capacity

We prove two bounds showing that if the eigenvalues of a matrix are clustered in a region of the complex plane then the corresponding discrete-time linear system requires significant energy to control. A curious feature of one of our bounds is that the dependence on the region is via its logarithmic capacity, which is a measure of how well a unit of mass may be spread out over the region to minimize a logarithmic potential.

Strong submultiplicativity of the Poincaré metric

We give a direct proof of an important result of Solynin which says that the Poincare metric is a strongly submultiplicative domain function. This result is then used to define a new capacity for compact subsets of the complex plane $${\mathbb {C}}$$ , which might be called Poincare capacity. If the compact set $$K \subseteq {\mathbb {C}}$$ is connected, then the Poincare capacity of K is the same as the logarithmic capacity of K. In this special case, the submultiplicativity is well-known and can be stated as an inequality for the normalized conformal map onto the complement of K. Using the connection between Poincare metrics and universal covering maps this inequality is extended to the much wider class of universal covering maps.

On the boundary-value problems for quasiconformal functions in the plane

Generalized solvability of the classical boundary-value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. Finally, some applications to the boundary-value problems for A-harmonic functions are given.

О задаче Римана–Гильберта для аналитических функций в круговых областях

It is proved the existence of single-valued analytic solutions in the unit disk and multivalent analytic solutions in domains bounded by a finite collection of circles for the Riemann-Hilbert problem with coefficients of sigma-finite variation and with boundary data that are measurable with respect to logarithmic capacity. It is shown that these spaces of solutions have the infinite dimension.

A converse to a theorem of Salem and Zygmund

By proving a converse to a theorem of Salem and Zygmund the paper gives a full description of the sets E of points x where the integral ∫01(F(x+t)−F(x−t))/tdt is infinite for a continuous and nondecreasing function F. It is shown that for this it is necessary and sufficient that E is a Gδ set of zero logarithmic capacity. Several corollaries are derived concerning boundary values of univalent functions.

Visualization of Buffer Capacity with 3-D “Topo” Surfaces: Buffer Ridges, Equivalence Point Canyons and Dilution Ramps

BufCap TOPOS is free software that generates 3-D topographical surfaces (“topos”) for acid–base equilibrium studies. It portrays pH and buffer capacity behavior during titration and dilution procedures. Topo surfaces are created by plotting computed pH and buffer capacity values above a composition grid with volume of NaOH as the x axis and overall system dilution as the y axis. What emerge are surface features that correspond to pH and buffer behaviors in aqueous solutions. Topo surfaces are created for pH, log buffer capacity, and linear buffer capacity. Equivalence point breaks become pH cliffs and logarithmic buffer capacity canyons that grow shallower with dilution. Areas of high buffer capacity become rounded ridges. Dilution alone generates 45° ramps. Example systems include hydrochloric acid, HCl (a strong acid); acetic acid, CH3COOH (a weak monoprotic acid); oxalic acid, HOOCCOOH (a weak diprotic acid); and l-glutamic acid hydrochloride, C5H9NO4·HCl (a weak triprotic acid). The Supporting Information includes a copy of the interactive BufCap TOPOS program, macro-enabled spreadsheets that quickly generate surfaces for any mono-, di-, or triprotic acid. Only acid dissociation constants, Ka values, need be changed. Other materials include teaching slides for lectures, plus worksheets and laboratory activities for first-year college courses and third-year or graduate analytical chemistry courses.

On the Riemann–Hilbert problem for the Beltrami equations in quasidisks

For the nondegenerate Beltrami equations in the quasidisks and, in particular, in smooth Jordan domains, we prove the existence of regular solutions of the Riemann–Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure (logarithmic capacity).

Inverse Polynomial Images are Always Sets of Minimal Logarithmic Capacity

In this paper, we prove that each inverse polynomial image (that is, each inverse image of an interval with respect to a polynomial mapping) is a set of minimal logarithmic capacity in a certain sense. Such sets play an important role in the theory of Pade-Approximation. The proofs are all based on the characterization theorems of Herbert Stahl.

Inequalities for harmonic measures with respect to non-overlapping domains

Let and be the components of the complement of a closed Jordan curve , and let , where . We extend the known inequality for the harmonic measures of with respect to and to the case of an arbitrary number of pairwise non- overlapping domains , , and prove analogous inequalities for the harmonic measures of sets concentrated in several discs or continua , , of a given logarithmic capacity. We also establish bounds for these measures in terms of the Schwarzian derivatives of functions that conformally map the domains onto the unit disc.