An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface. Partly supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. Partially supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation. Received: December 2, 2003; revised: January 18, 2004     

Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials

Summary. The paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and rates of convergence in a Sobolev space setting are given for the approximation with harmonic polynomials. Special attention is paid to the approximation of singular functions that arise typically in corners. These results for harmonic polynomials are extended to general elliptic equations with analytic coefficients by means of the theory of Bergman and Vekua; the approximation results for Laplace's equation hold true verbatim, if harmonic polynomials are replaced with generalized harmonic polynomials. The Partition of Unity Method is used in a numerical example to construct an operator adapted spectral method for Laplace's equation that is based on approximating with harmonic polynomials locally. Received May 26, 1997 / Revised version received September 21, 1998 / Published online September 7, 1999     

Generalized harmonic numbers with Riordan arrays

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain very simple connections between the Stirling numbers of both kinds and other generalized harmonic numbers. Further, we suggest that Riordan arrays associated with such generalized harmonic numbers allow us to find new generating functions of many combinatorial sums and many generalized harmonic number identities.     

Champagne subregions of the disk whose bubbles carry harmonic measure

We show that for any and any region G whose outer boundary equals , there is a sequence of pairwise disjoint closed disks in G such that is the set of accumulation points of and (harmonic measure on the boundary of for evaluation at some in ) is supported on . Received: 10 March 2000 / Revised version: 6 November 2001/Published online: 28 February 2002     

Boundary position feedback control of Kirchhoff's non‐linear strings

This paper is concerned with global stabilization of an undamped non‐linear string in the case where any velocity feedback is not available. The linearized system has an infinite number of poles and zeros on the imaginary axis. In the case where any velocity feedback is not available, a parallel compensator is effective. The stabilizer is constructed for the augmented system which consists of the controlled system and a parallel compensator. It is proved that the string can be stabilized by linear boundary control. Copyright © 2003 John Wiley & Sons, Ltd.     

Hypergeometric approach to Weideman’s conjecture

By applying the derivative operator to Dixon’s formula, we prove several harmonic number identities including one of the hardest challenge identities conjectured by Weideman (2003). Received: 28 October 2005     

On higher order Bessel potentials

The solutions of the equation in , where are investigated, Bessel potentials of higher order are defined, and recurrence relations between these solutions and these Bessel potentials are obtained. It is also proved that these solutions and the solutions of , under certain conditions, are identical. Received: 6 November 2002     

Instability for harmonic foliations on compact homogeneous spaces

In this paper we discuss the instability of harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces. We obtain a sufficient condition for a harmonic foliation to be unstable on compact submanifolds in a Euclidean space and on compact isotropy irreducible homogeneous spaces. We also classify compact symmetric spaces which have no non-trivial stable harmonic foliation.     

Gradient estimates for positive harmonic functions by stochastic analysis

We prove Cheng–Yau type inequalities for positive harmonic functions on Riemannian manifolds by using methods of Stochastic Analysis. Rather than evaluating an exact Bismut formula for the differential of a harmonic function, our method relies on a Bismut type inequality which is derived by an elementary integration by parts argument from an underlying submartingale. It is the monotonicity inherited in this submartingale which allows us to establish the pointwise estimates.     

The <Emphasis Type="Italic">p</Emphasis>-harmonic approximation and the regularity of <Emphasis Type="Italic">p</Emphasis>-harmonic maps

We extend to the degenerate case , Simons approach to the classical regularity theory of harmonic maps of Schoen & Uhlenbeck, by proving a p-Harmonic Approximation Lemma. This allows to approximate functions with p-harmonic functions in the same way as the classical harmonic approximation lemma (going back to De Giorgi) does via harmonic functions. Finally, we show how to combine this tool with suitable regularity estimates for solutions to degenerate elliptic systems with a critical growth right hand side, in order to obtain partial -regularity of p-harmonic maps.Received: 2 November 2002, Accepted: 10 July 2003, Published online: 4 September 2003Mathematics Subject Classification (2000): 35J70, 49N60, 49Q60     

Harmonic maps and asymptotic Teichmüller space

In this paper, the asymptotic boundary behavior of a Hopf differential or the Beltrami coefficient of a harmonic map is investigated and certain compact properties of harmonic maps are established. It is shown that, if f is a quasiconformal harmonic diffeomorphism between two Riemann surfaces and is homotopic to an asymptotically conformal map modulo boundary, then f is asymptotically conformal itself. In addition, we prove that the harmonic embedding map from the Bers space B Q D (X) of an arbitrary hyperbolic Riemann surface X to the Teichmüller space T (X) induces an embedding map from the asymptotic Bers space A B Q D (X), a quotient space of B Q D (X), into the asymptotic Teichmüller space AT (X). The work was supported by a Foundation for the Author of National Excellent Doctoral Dissertation (Grant No. 200518) of PR China and the National Natural Science Foundation of China (Grant No. 10401036).     

Regularization of convex variational problems with applications to generalized Newtonian fluids

We study variational problems with convex integrands of very general structure by introducing certain regularizations leading to particular minimizers. In a second part we apply the method to stationary generalized Newtonian fluids which gives the existence of solutions under weak hypotheses on the dissipative potential. In particular the potential is not required to be a strictly convex function.Received: 29 July 2003     

Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

A constructive method for starlike harmonic mappings

Summary The idea initiated by Opfer for constructing conformal mappings from the disk onto starlike domains is generalized for univalent harmonic mappings. This is of some interest, since such mappings are not characterized by analytic means.This work was supported in partsby a Promotion of Research Grant from the TECHNION, Haifaby an Undergraduate Student Research A ward from the NSERCby grants from the NSERC and the FCAR