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.

Logarithmic potential and generalized analytic functions

The study of the Dirichlet problem in the unit disk $\mathbb D$ with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua \cite{Ve} has been devoted to boundary-value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb C$. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar\'{e} problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Symmetry results for variational energies on convex polygons

We prove that, in the class of convex polygons with a given number of sides, the regular $n$-gon is optimal for some shape optimization problems involving the torsional rigidity, the principal frequency of the Laplacian, or the logarithmic capacity.

On boundary-value problems for generalized analytic and harmonic functions

The present paper is a natural continuation of our last articles on the Riemann, Hilbert, Dirichlet, Poincaré, and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic functions and the so-called A-harmonic functions with arbitrary boundary data that are measurable with respect to the logarithmic capacity. Here, we extend the corresponding results to generalized analytic functions h : D→C with sources g : ∂z-h = g ∈ Lp , p > 2, and to generalized harmonic functions U with sources G : ΔU =G ∈Lp , p > 2. Our approach is based on the geometric (functional-theoretic) interpretation of boundary values in comparison with the classical operator approach in PDE. Here, we will establish the corresponding existence theorems for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations ΔU =G with arbitrary boundary data that are measurable with respect to the logarithmic capacity. A few mixed boundary-value problems are considered as well. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.