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1. Malmheden's theorem revisitedAgranovsky, M.

et al.

Khavinson, D.

Shapiro, Harold

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Malmheden's theorem revisited2010In: Expositiones mathematicae, ISSN 0723-0869, E-ISSN 1878-0792, Vol. 28, no 4, p. 337-350Article in journal (Refereed)

Abstract [en]

In 1934 Malmheden [16] discovered an elegant geometric algorithm for solving the Dirichlet problem in a ball. Although his result was rediscovered independently by Duffin (1957) [8] 23 years later, it still does not seem to be widely known. In this paper we return to Malmheden's theorem, give an alternative proof of the result that allows generalization to polyharmonic functions and, also, discuss applications of his theorem to geometric properties of harmonic measures in balls in R-n.

We study the minimization problem for the Dirichlet integral in some standard classes of analytic functions. In particular, we solve the minimal area a(2)-problern for convex functions and for typically real functions. The latter gives a new solution to the minimal area a(2)-problem for the class S of normalized univalent functions in the unit disc.

We consider the Dirichlet problem for the Laplace operator with rational data on the boundary of a planar domain. Our main results include a characterization of the disk as the only domain for which all solutions are rational and a characterization of the simply connected quadrature domains as the only ones for which all solutions are algebraic of a certain type.

Univ Calif San Diego, Dept Math, La Jolla, CA 92093 USA..

Khavinson, D.

Univ S Florida, Dept Math & Stat, Tampa, FL 33620 USA..

Shapiro, Harold S.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Two-dimensional Shapes and Lemniscates2011In: COMPLEX ANALYSIS AND DYNAMICAL SYSTEMS IV, PT 1: FUNCTION THEORY AND OPTIMIZATION / [ed] Agranovsky, M BenArtzi, M Galloway, G Karp, L Reich, S Shoikhet, D Weinstein, G Zalcman, L, AMER MATHEMATICAL SOC , 2011, p. 45-+Conference paper (Refereed)

Abstract [en]

According to the theorem of A. Kirillov, every smooth, closed Jordan curve in the plane can be represented by its "fingerprint", a diffeomorphism of the unit circle. In this paper we show that the fingerprints of (polynomial) lemniscates of degree n are given by n-th roots of Blaschke products of degree re. The latter are dense in the space of all diffeomorphisms of the unit circle. Moreover, we also prove that every diffeomorphism of the unit circle induced by the n-th root of a Blaschke product with n zeros fingerprints a polynomial lernniscate of the same degree.

5. What is a quadrature domain?

Gustafsson, Björn

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Shapiro, Harold S.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

What is a quadrature domain?2005In: Quadrature Domains and Their Applications: The Harold S. Shapiro Anniversary Volume / [ed] Ebenfelt, P; Gaustafsson, B; Khavinson, D; Putinar, M, 2005, Vol. 156, p. 1-25Conference paper (Refereed)

Abstract [en]

We give an overview of the theory of quadrature domains with indications of some if its ramifications.

6. Poincare's variational problem in potential theoryKhavinson, Dmitry

et al.

Putinar, Mihai

Shapiro, Harold S.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

One of the earliest attempts to rigorously prove the solvability of Dirichlet's boundary value problem was based on seeking the solution in the form of a "potential of double layer", and this leads to an integral equation whose kernel is ( in general) both singular and non-symmetric. C. Neumann succeeded with this approach for smoothly bounded convex domains, and H. Poincare, by a tremendous tour de force, showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity. In this work he was ( according to his own account) guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain, more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain. Unfortunately, a rigorous treatment of this problem was not possible with the tools available at that time ( as Poincare was well aware). So far as we know, the only one to propose a rigorous treatment of Poincare's problem was T. Carleman ( in the two-dimensional case) in his doctoral dissertation. Thanks to later developments ( especially concerning Sobolev spaces, and spectral theory of operators on Hilbert space) we can now give a complete, general and rigorous account of Poincare's variational problem, and that is the main object of the present paper. As a by-product, we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensions the basic properties of the analogue of the planar Bergman - Schiffer singular integral equation. We interpret Poincare's variational principle as a non-selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials. We also prove a series of novel spectral analysis facts ( some of them conjectured by Poincare) related to the Poincare - Neumann integral operator. One of the earliest attempts to rigorously prove the solvability of Dirichlet's boundary value problem was based on seeking the solution in the form of a "potential of double layer", and this leads to an integral equation whose kernel is (in general) both singular and non-symmetric. C. Neumann succeeded with this approach for smoothly bounded convex domains, and H. Poincare, by a tremendous tour de force, showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity. In this work he was (according to his own account) guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain, more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain. Unfortunately, a rigorous treatment of this problem was not possible with the tools available at that time (as Poincare was well aware). So far as we know, the only one to propose a rigorous treatment of Poincare's problem was T. Carleman (in the two-dimensional case) in his doctoral dissertation. Thanks to later developments (especially concerning Sobolev spaces, and spectral theory of operators on Hilbert space) we can now give a complete, general and rigorous account of Poincare's variational problem, and that is the main object of the present paper. As a by-product, we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensins the basic properties of the analogue of the planar Bergman-Schiffer singular integral equation. We interpret Poincare's variational principle as a non-selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials. We also prove a series of novel spectral analysis facts (some of them conjectured by Poincare) related to the Poincare-Neumann integral operator.

City Univ Hong Kong, Dept Math, Hong Kong, Hong Kong, Peoples R China..

Zeilberger, Doron

Rutgers State Univ, Dept Math, Piscataway, NJ 08855 USA..

Xu, Yuan

Univ Oregon, Dept Math, Eugene, OR 97403 USA..

Donald J. Newman (1930-2007) In Memoriam2008In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 154, no 1, p. 37-58Article in journal (Refereed)