Isoperimetric inequalities for L p -norms of the stress function of a multiply connected plane domain

Let u(x,G) be the stress function of a multiply connected plane domain G. We construct new domain functionals depending on this stress function which are isoperimetrically monotone with respect to the free parameter. A particular case of the proved result is the Payne inequality for the torsional rigidity of G.     

Integral properties of the classical warping function of a simply connected domain

Let u(z,G) be the classical warping function of a simply connected domain G. We prove that the L ^p -norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = sup _x∈G u(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the L ^p -norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne.     

Isoperimetric properties of Euclidean boundary moments of a simply connected domain

We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For L ^p -norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk.     

How geodesics approach the boundary in a simply connected domain

This work was supported by NSF Grant DMS 92-04092 and an Alfred P. Sloan Research Fellowship.     

Integral representations of generalized axially symmetric potentials in a simply connected domain

We obtain integral representations of generalized axially symmetric potentials via analytic functions of a complex variable that are defined in an arbitrary simply connected bounded domain symmetric with respect to the real axis. We prove that these integral representations establish a one-to-one correspondence between analytic functions of a complex variable that take real values on the real axis and generalized axially symmetric potentials of certain classes.     

On the approximate conformal mapping of the unit disk on a simply connected domain

We present a method of constructing the analytic function realizing an approximate conformal mapping of the unit disk on an arbitrary simply connected domain with the given smooth parametrically defined boundary. The method is based on a new boundary parameterization. Solution of the problem is reduced to the Fredholm integral equation of the second kind. We present the examples of three ways to solve the integral equation.     

Estimation of the L p -norms of stress functions for finitely connected plane domains

Let u(x, G) be the classical stress function of a finitely connected plane domain G. The isoperimetric properties of the L ^p -norms of u(x, G) are studied. Payne’s inequality for simply connected domains is generalized to finitely connected domains. It is proved that the L ^p -norms of the functions u(x, G) and u ^?1 (x, G) strictly decrease with respect to the parameter p, and a sharp bound for the rate of decrease of the L ^p -norms of these functions in terms of the corresponding L ^p -norms of the stress function for an annulus is obtained. A new integral inequality for the L ^p -norms of u(x, G), which is an analog of the inequality obtained by F. G. Avkhadiev and the author for the L ^p -norm of conformal radii, is proved.     

Invariant ordering on the simply connected covering of the Shilov boundary of a symmetric domain

The Shilov boundary of a symmetric domain D = G/K of tube type has the form G/P, where P is a maximal parabolic subgroup of the group G. We prove that the simply connected covering of the Shilov boundary possesses a unique (up to inversion) invariant ordering, which is induced by the continuous invariant ordering on the simply connected covering of G and can readily be described in terms of the corresponding Jordan algebra.     

Non-uniqueness of the reconstruction for connected and simply connected sets in the plane by their fixed finite projections

We discuss a problem to reconstruct the measurable sets in the plane from their fixed finite projections. In the main theorem, we construct an example of connected and simply connected polygons which are not uniquely reconstructed by their fixed finite projections. We also make a comparison between our main theorem and the known results on this problem.     

Mappings of the sphere to a simply connected space

Fix an m ∈ ℕ, m ≥ 2. Let Y be a simply connected pointed CW-complex, and let B be a finite set of continuous mappings a: S^m → Y respecting the distinguished points. Let Γ(a) ⊂ S^m × Y be the graph of a, and we denote by [a] ∈ π_m(Y) the homotopy class of a. Then for some r ∈ ℕ depending on m only, there exist a finite set E ⊂ S^m × Y and a mapping k: E(r) = {F ⊂ E: |F| ≤ r} → π_m(Y) such that for each a ∈ B we have

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces E _q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W ^r E _p(Ω) and W ^r E _p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.     

The calculation of the L 2-norm of the index of a plane curve and related formulas

We provide formulas for calculating the L ²-norm of the index function of a rectifiable closed curve in the complex plane. Some applications to isoperimetric inequalities are given. The main tool used is the decomposition of a rectifiable closed curve into a sequence of Jordan curves, curves with null index functions, and an exceptional set.     

Completing the conformal boundary of a simply connected Lorentz surface

This paper extends Kulkarni's conformal boundary for a simply connected Lorentz surface to a compact conformal boundary . The procedure used is analogous to Carathéodory's construction (in the definite metric setting) of prime ends from the accessible points of a bounded simply connected planar domain. The space of conformal boundary elements is homeomorphic to the circle, and contains Kulkarni's conformal boundary as a dense subspace.

     

Decoupling of the quasistatic system of thermoelasticity with Riemann problems on the bounded simply connected domain

Under the displacement and stress satisfying Riemann boundary value condition, the decoupled quasistatic linear thermoelasticity system is discussed on bounded simply connected domain. The quasistatic equilibrium equation is solved by using Riemann boundary value problem theory. Also decoupled temperature equation is studied by applying the contractive mapping principle. Finally, existence and analyticity of the solution are proved.