The Liouville equation in an annulus

We give the solutions to the Liouville equation in an annulus A of R² that satisfy a certain Neumann condition on each component of ∂A. As a consequence, we classify all the metrics of constant curvature in A that have constant geodesic curvature on ∂A.     

Korn's inequality and Donati's theorem for the conformal Killing operator on pseudo-Euclidean space

We prove the Korn's inequality for the conformal Killing operator on pseudo-Euclidean space Rp,q, and an existence theorem for solutions to the non-homogeneous conformal Killing equation, which is a pseudo-Euclidean conformal generalization of Donati's theorem for Euclidean Killing operator.     

Estimates for conformal capacity

Let a,b,c,d be distinct points on overline bf R ⁿ . By p we denote the minimal conformal capacity of all rings (E,F) with a,b ∈ E and c,d∈ F . For n=2 , we use explicit expressions of p in terms of complete elliptic integrals to prove a sharp inequality that connects p and the conformal capacity of Teichmüller's ring. We also show, by a concrete example, how we can use techniques involving polarization and hyperbolic geometry to prove estimates for the conformal capacity of rings. November 16, 1998. Date revised: March 2, 1999. Date accepted: May 7, 1999.     

On the theory and application of a domain decomposition method for computing conformal modules

We consider the theory and application of a domain decomposition method for computing the conformal modules of long quadrilaterals. The method has been studied already by us and also by Gaier and Hayman. Our main purpose here is to extend its area of application and, in the same time, improve some of our earlier error estimates.     

A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities

We prove several identities relating three-variable Mahler measures to integrals of inverse trigonometric functions. After deriving closed forms for most of these integrals, we obtain ten explicit formulas for three-variable Mahler measures. Several of these results generalize formulas due to Condon and Lalín. As a corollary, we also obtain three q-series expansions for the dilogarithm.     

GJMS operators and Q-curvature for conformal Codazzi structures

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersurfaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.     

二阶椭圆型方程奇摄动边值问题的渐近解

研究了一类二阶线性椭圆型方程的奇摄动边值问题.利用边界层函数法构造出问题的零次形式近似,并应用椭圆型算子的最大值原理对问题的解作出渐近估计.     

Elliptic equations for measures: Regularity and global bounds of densities

We consider elliptic equations of the form L^*μ=ν for measures on . The membership of solutions in the Sobolev classes is established under appropriate conditions on the coefficients of L. Bounds of the form (x)CΦ(x)⁻¹ for the corresponding densities are obtained.     

Periodic reiterated homogenization for elliptic functions

In this paper, we study reiterated homogenization for equations of the form . We assume that a is a Carathéodory function and satisfies some monotonicity and growth conditions and its reiterated unfolding converges almost everywhere to a Carathéodory type function. Under these assumptions, we show that the sequence of solutions converges to the solution of a limit variational problem. In particular this contains the case , where a is periodic in the second and third arguments, and continuous in each argument.     

On nonuniqueness of solutions to elliptic equations for probability measures

We consider the set of all probability measures μ on Rd satisfying an elliptic equation L^∗μ=0 in the weak worm. We give sufficient conditions in order that this set contains at least two different elements. We also construct new examples of nonuniqueness for such equations.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps. The authors are supported in part by an HKRGC grant.     

Asymptotic coefficients for Gaussian radial basis function interpolants

Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by ?(x;α,h)≡exp(-[α²/h²]x²). The only significant numerical parameter is α, the inverse width of the RBF functions relative to h. In the limit α→0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(π²/[4α²]). However, we also show that the approximation to the constant f(x)≡1 is a Jacobian theta function whose coefficients do not blow up as α→0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter α are analyzed. For α≈1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(10⁴) and the error saturation is smaller than machine epsilon, so this α is the center of a “safe operating range” for Gaussian RBFs.     

On conformal invariants for elliptic systems with multiple critical exponents

In this article we study multiple solutions of the prescribed curvatures problem for a compact Riemannian manifold with smooth boundary and with indefinite signs of the energy function and nonlinearities. We generalize the conception of the critical Palais–Smale level and find new type of conformally invariant necessary and sufficient conditions for existence of multiple solutions of the problem. The second author was supported in part by grants INTAS 03-51-5007, RFBR 05-01-00370, 05-01-00515.     

On conformal measures for infinitely renormalizable quadratic polynomials

We study a conformal measure for an infinitely renormalizable quadratic poly- nomial.We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds.The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.     

Asymptotic paths for subsolutions of quasilinear elliptic equations

Letu be an entire lower semicontinuous subsolution to the quasilinear elliptic equation divA(x,u)=0 in ⁿ . It is shown that ifu is not bounded above, then there exists a path going to infinity along whichu tends to infinity. The result extends works of Talpur, Fuglede, and others. Growth aspects of subsolutions are also studied.     

Multi-layer solutions for an elliptic problem

In this paper, we prove the existence of solutions with multiple interior layers for an elliptic Neumann problem.     

Adaptive search with stochastic acceptance probabilities for global optimization

We present an extension of continuous domain Simulated Annealing. Our algorithm employs a globally reaching candidate generator, adaptive stochastic acceptance probabilities, and converges in probability to the optimal value. An application to simulation-optimization problems with asymptotically diminishing errors is presented. Numerical results on a noisy protein-folding problem are included.     

Integral representations for elliptic functions

We derive new integral representations for constituents of the classical theory of elliptic functions: the Eisenstein series, and Weierstrass' ℘ and ζ functions. The derivations proceed from the Laplace-Mellin representation of multipoles, and an elementary lemma on the summation of 2D geometric series. In addition, we present results concerning the analytic continuation of the Eisenstein series to an entire function in the complex plane, and the value of the conditionally convergent series, denoted by below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio.¹