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  • 1. Bell, Steven
    et al.
    Gustafsson, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sylvan, Zachary
    Szegö coordinates, quadrature domains, and double quadrature domains2011In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 11, no 1, p. 25-44Article in journal (Refereed)
    Abstract [en]

    We define Szegö coordinates on a finitely connected smoothly bounded planar domain which effect a holomorphic change of coordinates on the domain that can be as close to the identity as desired and which convert the domain to a quadrature domain with respect to boundary arc length. When these Szegö coordinates coincide with Bergman coordinates, the result is a double quadrature domain with respect to both area and arc length. We enumerate a host of interesting and useful properties that such double quadrature domains possess, and we show that such domains are in fact dense in the realm of bounded C-smooth finitely connected domains.

  • 2.
    Ebenfelt, Peter
    et al.
    Mathematics, University of California in San Diego.
    Gustafsson, BjörnKTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).Khavinson, DmitryMathematics, University of Arkansas.Putinar, MihaiMathematics, University of California in Santa Barbara.
    Quadrature Domains and Their Applications: The Harold S. Shapiro Anniversary Volume2005Conference proceedings (editor) (Refereed)
  • 3. Gaudiello, A.
    et al.
    Gustafsson, Björn
    KTH, Superseded Departments, Mathematics.
    Lefter, C.
    Mossino, J.
    Coupled and uncoupled limits for a N-dimensional multidomain Neumann problem2000In: Comptes rendus de l'Académie des sciences. Série 1, Mathématique, ISSN 0764-4442, E-ISSN 1778-3577, Vol. 330, no 11, p. 985-990Article in journal (Refereed)
    Abstract [en]

    We consider the quasilinear Neumann problem with exponent p, in a multi-domain of R-N made of the union of a cylinder with given height and small cross section and a cylinder with small height and given cross section. Assuming that the volumes of the two cylinders tend to zero with same order, we prove that the limit problem is posed in the union of the limit domains, with respective dimensions 1 and (N - 1). Moreover this limit problem is coupled if p > N - 1 and uncoupled otherwise.

  • 4. Gaudiello, A.
    et al.
    Gustafsson, Björn
    KTH, Superseded Departments, Mathematics.
    Lefter, C.
    Mossino, J.
    Gustafsson, Björn2002In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 15, no 2, p. 181-201Article in journal (Refereed)
    Abstract [en]

    We consider a quasilinear Neumann problem with exponent p is an element of]1, +infinity[, in a multidomain of R-N, N greater than or equal to 2, consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the other one with small height and given cross section. Assuming that the volumes of the two cylinders tend to zero with same rate, we prove that the limit problem is well posed in the union of the limit domains, with respective dimension 1 and N - 1. Moreover, this limit problem is coupled if p > N - 1 and uncoupled if 1 < p less than or equal to N - 1.

  • 5.
    Gustafsson, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Exponential transforms, resultants and moments2014In: Harmonic and Complex Analysis and its Applications / [ed] Alexander Vasilev, Switzerland: Springer, 2014, p. 287-323Chapter in book (Refereed)
    Abstract [en]

    We give an overview of some recent developments concerning harmonic and other moments of plane domains, their relationship to the Cauchy and exponential transforms, and to the meromorphic resultant and elimination function. The paper also connects to certain topics in mathematical physics, for example domain deformations generated by harmonic gradients (Laplacian growth) and related integrable structures.

  • 6.
    Gustafsson, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lectures on balayage2004In: Univ. Joensuu Dept. Rep. Ser.Article, review/survey (Refereed)
  • 7.
    Gustafsson, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The string equation for polynomials2018In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 8, no 4, p. 637-653Article in journal (Refereed)
    Abstract [en]

    For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova–Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we show that the string equation makes sense and holds for general polynomials.

  • 8.
    Gustafsson, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Vortex motion and geometric function theory: the role of connections2019In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 377, no 2158, article id 20180341Article in journal (Refereed)
    Abstract [en]

    We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi-Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections. This article is part of the theme issue 'Topological and geometrical aspects ofmass and vortex dynamics'.

  • 9.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    He, C. Y.
    Milanfar, P.
    Putinar, M.
    Reconstructing planar domains from their moments2000In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 16, no 4, p. 1053-1070Article in journal (Refereed)
    Abstract [en]

    In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of a class of planar semi-analytic domains from their moments. A part of this class, known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the complex plane, and is therefore of great practical importance. We provide an exact reconstruction algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will be presented. In addition, relations of the present theory to computer-assisted tomography and a geophysical inverse problem will be briefly discussed.

  • 10.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Heron, B.
    Mossino, J.
    Gamma-convergence of stratified media with measure-valued limits2000In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 22, no 04-mar, p. 261-302Article in journal (Refereed)
    Abstract [en]

    We consider energy functionals, or Dirichlet forms, [GRAPHICS] for a class G of bounded domains Omega subset of R-N, with epsilon>0 a fine structure parameter and with symmetric conductivity matrices A(epsilon) = (a(ij)(epsilon)) is an element of L-loc(infinity)(R)(NxN) which are functions only of the first coordinate x(1) and which are locally uniformly elliptic for each fixed epsilon>0. We show that if the functions (of x(1)) b(11)(epsilon) = 1/a(11)(epsilon), b(1j)(epsilon) = a(1j)(epsilon)/a(11)(epsilon) (j greater than or equal to 2), b(ij)(epsilon) = a(ij)(epsilon) - a(i1)(epsilon)a(1j)(epsilon)/ a(11)(epsilon) (i, j greater than or equal to 2) converge weakly* as measures towards corresponding limit measures b(ij) as epsilon --> 0, if the (1,1)-coefficient m(11)(epsilon) of (A(epsilon))(-1) is bounded in L-loc(1)(R) and if none of its weak* cluster measures has atoms in common with b(ii), i greater than or equal to 2, then the family J(epsilon) = {J(Omega)(epsilon)}(Omega is an element of g) Gamma-converges in a local sense towards a naturally defined limit family J = {J(Omega))(Omega is an element of G) as epsilon-->0. An alternative way of formulating the conclusion is to say that the energy densities (A(epsilon)del u,del u) Gamma-converge in a distributional sense towards the corresponding limit density. Writing J(Omega)(epsilon) in terms of B-epsilon = (b(ij)(epsilon)) it becomes [GRAPHICS] and the definition of J(Omega) and the limit density (A del u, del u) is obtained by properly replacing the b(ij)(epsilon) is an element of L-loc(infinity)(R) by the limit measures b(ij) and making sense to everything for u in a certain linear subspace of L-loc(2)(R-N).

  • 11.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lin, Yu-Lin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the dynamics of roots and poles for solutions of the polubarinova-galin equation2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 259-286Article in journal (Refereed)
    Abstract [en]

    We study the dynamics of roots of f'(zeta, t), where F(zeta, t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give examples of the sometimes complicated motion of roots, but show also that the asymptotic behavior is simple. More generally we allow f'(zeta, t) to be a rational function and give sharp estimates for the motion of poles and for the decay of the Taylor coefficients. We also prove that any global in time locally univalent solution actually has to be univalent.

  • 12.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Mossino, J.
    Compensated compactness for homogenization and reduction of dimension: The case of elastic laminates2006In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 47, no 02-jan, p. 139-169Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to extend, to the linear elasticity system, the asymptotic analysis by compensated compactness previously developed by the authors for the linear diffusion equation. For simplicity, we restrict ourselves to stratified media. In the case of sole homogenization we recover the classical result of W.H. Mc Connel, deriving explicitly the effective elasticity tensor for stratified media. Here we give a new proof of his result, based on compensated compactness and on a technique of decomposing matrices. As for the case of simultaneous homogenization and reduction of dimension, we perform the asymptotic analysis, as the thickness tends to zero, of a three-dimensional laminated thin plate having an anisotropic, rapidly oscillating elasticity tensor. The limit problem is presented in three different ways, the final formulation being a fourth-order problem on the two-dimensional plate, with explicitly given elasticity tensors and effective source terms.

  • 13.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Mossino, J.
    Explicit limits for nonperiodic homogenization and reduction of dimension2002In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 334, no 11, p. 977-982Article in journal (Refereed)
    Abstract [en]

    The aim of this Note is to give explicit limit expressions. for diffusion equations involving a small parameter E, describing both nonperiodic homogenization and reduction of dimension. We consider two kinds of reduction of dimension: the case of plates and the case of thin cylinders. In particular, we give the limit diffusion equation for stratified plates. This is completely explicit and requires no special assumption, except stratification. In the case of thin cylinders, the formulae are less explicit, but we also indicate some simple applications to fibered materials.

  • 14.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Mossino, J.
    Non-periodic explicit homogenization and reduction of dimension: the linear case2003In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 68, no 3, p. 269-298Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to give explicit limit expressions, for diffusion equations involving a small parameter epsilon, describing both nonperiodic homogenization and reduction of dimension. In other words, we give the limit behaviour, when epsilon tends to zero, of the diffusion equation in a thin domain, with thickness of order epsilon, when the coefficients of the equation also depend on epsilon and may present rapid, nonperiodic oscillations, provided they satisfy a suitable compensated compactness condition. We consider two kinds of reduction of dimension: the case of thin plates (3D --> 2D) and the case of thin cylinders (3D --> 1D). In particular, we give the limit diffusion equation for laminated plates. This is completely explicit and requires no special assumption, except stratification. In the case of thin cylinders, the formulae are less explicit, but we also indicate some simple applications.

  • 15.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Mossino, Jacqueline
    A criterion for H-convergence in elasticity2007In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 51, no 3-4, p. 247-269Article in journal (Refereed)
    Abstract [en]

    We give a criterion for H-convergence of elasticity tensors in terms of ordinary weak convergence of the factors in certain quotient representations of the tensors.

  • 16.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Prokhorov, D.
    Vasil'ev, A.
    Infinite lifetime for the starlike dynamics in Hele-Shaw cells2004In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 132, no 9, p. 2661-2669Article in journal (Refereed)
    Abstract [en]

    One of the folklore questions in the theory of free boundary problems is the lifetime of the starlike dynamics in a Hele-Shaw cell. We prove precisely that, starting with a starlike analytic phase domain Omega(0), the Hele-Shaw chain of subordinating domains Omega( t), Omega(0) = Omega(0), exists for an infinite time under injection at the point of starlikeness.

  • 17.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, M.
    Linear analysis of quadrature domains - IV2005In: QUADRATURE DOMAINS AND THEIR APPLICATIONS: THE HAROLD S. SHAPIRO ANNIVERSARY VOLUME / [ed] Ebenfelt, P; Gaustafsson, B; Khavinson, D; Putinar, M, 2005, Vol. 156, p. 173-194Conference paper (Refereed)
    Abstract [en]

    The positive definiteness of the exponential transform of a planar domain is proved by elementary means. This direct approach avoids the heavy machinery of the theory of hyponormal operators and leads to a better understanding of the linear data associated in previous works to a quadrature domain.

  • 18.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Putinar, M.
    Linear analysis of quadrature domains. II2000In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 119, p. 187-216Article in journal (Refereed)
    Abstract [en]

    The natural correspondence between bounded planar quadrature domains, in the terminology of Aharonov-Shapiro, and certain square matrices with a distinguished cyclic vector is further exploited. Two different cubature formulas on quadrature domains, that is the computation of the integral of a real polynomial, are presented. The minimal defining polynomial of a quadrature domain is decomposed uniquely into a linear combination of moduli squares of complex polynomials. The geometry of a canonical rational embedding of a quadrature domain into the projective complement of a real affine ball is also investigated. Explicit computations on order-two quadrature domains illustrate the main results.

  • 19.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Putinar, M.
    On exact quadrature formulas for harmonic functions on polyhedra2000In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 128, no 5, p. 1427-1432Article in journal (Refereed)
    Abstract [en]

    A classical quadrature result for analytic functions of a complex variable due to Motzkin and Schoenberg is extended to higher dimensions. A general scheme for integrating on polyhedra solutions of partial differential equations is discussed.

  • 20.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Putinar, M.
    The exponential transform: A renormalized Riesz potential at critical exponent2003In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 52, no 3, p. 527-568Article in journal (Refereed)
    Abstract [en]

    For an arbitrary domain in R-n we consider the exponential of a suitably normalized Riesz potential of first nonintegrable index. This gives a positive function, with certain monotonicity properties, defined in the complement of the domain, and vanishing on its boundary. In two dimensions this function, polarized into two complex variables, has previously been studied and proved to be useful within operator theory, moment problems and other problems of domain identification, and for proving regularity of free boundaries. Even in the absence of a natural polarization there are noteworthy properties of the higher dimensional exponential transform. For instance we show that this function is superharmonic and that it tends to zero at smooth points of the boundary, with slope bounded from above and below by constants depending solely on the local curvature. The analytic continuation configuration of the exponential transform of a convex polyhedron shows that it is indeed a natural defining function of its boundary. In addition, computations of this transform are carried out in full detail for some domains bounded by lower degree algebraic surfaces. There is also an inner exponential transform, defined inside the domain. If the domain is convex then the ordinary Newtonian potential of it can, for points inside the domain, be interpreted is an arithmetic meanvalue with respect to the solid angle of the squared distance to the boundary. The interior exponential transform is then the corresponding geometric meanvalue.

  • 21.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Putinar, M.
    Shapiro, H. S.
    Restriction operators, balayage and doubly orthogonal systems of analytic functions2003In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 199, no 2, p. 332-378Article in journal (Refereed)
    Abstract [en]

    Systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by Walsh and Szego, and in full generality by Bergman. In principle, these are very interesting objects, allowing application to analytic continuation that is not restricted (as Weierstrassian continuation via power series) either by circular geometry or considerations of locality. However, few explicit examples are known, and in general one does not know even gross qualitative features of such systems. The main contribution of the present paper is to prove qualitative results in a quite general situation. It is by now very well known that the phenomenon of double orthogonality is not restricted to Bergman spaces of analytic functions, nor even indeed has it any intrinsic relation to analyticity; its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a Hilbert space of functions on some set, to a subset thereof, provided this restriction is injective and compact. However, in this paper only Hilbert spaces of analytic functions are considered, especially Bergman spaces. In the case of the Hardy spaces Fisher and Micchelli discovered remarkable qualitative features of doubly orthogonal systems, and we have shown how, based on the classical potential-theoretic notion of balayage, and its modern generalizations, one can deduce analogous results in the Bergman space set-up, but with restrictions imposed on the geometry of the considered domains and measures; these were not needed in the Fisher-Micchelli analysis, but are necessary here as shown by examples. From a more constructive point of view we study the Bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible (as an inversion followed by a logarithmic derivative) in terms of the polynomial equation of the boundary of the inner domain.

  • 22.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, Mihai
    Univ Calif Santa Barbara, Dept Math, Santa Barbara, CA 93106 USA.;Newcastle Univ, Sch Math Stat & Phys, Newcastle Upon Tyne NE1 7RU, Tyne & Wear, England..
    A field theoretic operator model and Cowen-Douglas class2019In: Banach Journal of Mathematical Analysis, ISSN 1735-8787, Vol. 13, no 2, p. 338-358Article in journal (Refereed)
    Abstract [en]

    In resonance with a recent geometric framework proposed by Douglas and Yang, we develop a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space. By taking advantage of the refined existing theory of the principal function of a hyponormal operator, we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. We propose a natural field theory interpretation of the resulting resolvent functional model.

  • 23.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, Mihai
    Analytic continuation of the exponential transform from convex cavities2007In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 328, no 2, p. 995-1006Article in journal (Refereed)
    Abstract [en]

    The analytic continuation of the exponential transform of a domain in R-n is proved under some global geometric assumptions on the boundary. Two approximation schemes of the continued transform (one based on a Taylor series truncation, the other on a global eigenfunction expansion) are also discussed.

  • 24.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Putinar, Mihai
    Univ Calif Santa Barbara, Dept Math, Santa Barbara, CA 93106 USA.;Newcastle Univ, Sch Math Stat & Phys, Newcastle Upon Tyne NE1 7RU, Tyne & Wear, England..
    Finite term relations for the exponential orthogonal polynomials*2020In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 15, article id 5Article in journal (Refereed)
    Abstract [en]

    The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function g supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthogonal polynomials. On function theory side, we offer an effective way based on the Cauchy transforms of to decide whether a (d + 2)-term relation among the exponential orthogonal polynomials exists; in that case we indicate how the shade function g can be reconstructed from a resulting polynomial of degree d and the Cauchy transform of g. A discussion of the relevance of the main concepts in Hele-Shaw dynamics completes the article.

  • 25.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, Mihai
    UCSB, Dept Math, Santa Barbara, CA 93106 USA.;Newcastle Univ, Sch Math Stat & Phys, Newcastle Upon Tyne NE1 7RU, Tyne & Wear, England..
    Line bundles defined by the Schwarz function2018In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 8, no 2, p. 171-183Article in journal (Refereed)
    Abstract [en]

    Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.

  • 26.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, Mihai
    Selected topics on quadrature domains2007In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 235, no 02-jan, p. 90-100Article in journal (Refereed)
    Abstract [en]

    This is a selection of facts, old and recent, about quadrature domains. The text, written in the form of a survey, is addressed to non-experts and covers a variety of phenomena related to quadrature domains, such as: the difference between quadrature domains for subharmonic, harmonic and respectively complex analytic functions, geometric properties of the boundary, instability in the reverse Hele-Shaw flow, dependence and nonuniqueness on the quadrature data, interpretation in terms of function theory on Riemann surfaces, a matrix model and a reconstruction algorithm. Also there are some low degree/order examples where computations can be carried out in detail.

  • 27.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, Mihai
    Saff, Edward B.
    Stylianopoulos, Nikos
    Bergman orthogonal polynomials on an archipelago2008In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 346, no 9-10, p. 499-502Article in journal (Refereed)
    Abstract [fr]

    Growth estimates for orthogonal polynomials with respect to area measure (Bergman polynomials) over the union of finitely many Jordan regions with piecewise smooth boundary are obtained by a careful investigation of the Green function of the complement, and of Schwarz reflection in analytic arcs of the boundary. As applications we obtain a detailed picture of the limiting zero distribution of Bergman's orthogonal polynomials, and also we propose a robust reconstruction algorithm of the original open set, starting from incomplete data (such as obtained by geometric tomography).

  • 28.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, Mihai
    Saff, Edward B.
    Stylianopoulos, Nikos
    Bergman polynomials on an archipelago: Estimates, zeros and shape reconstruction2009In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 222, no 4, p. 1405-1460Article in journal (Refereed)
    Abstract [en]

    Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains ( called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago vertical bar z(m)-1 vertical bar < r(m), 0 < r < 1, which consists of m islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szego orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.

  • 29.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Sakai, M.
    On the curvature of the free boundary for the obstacle problem in two dimensions2004In: Nonlinear Differential Equation Models / [ed] Jungel, A; Manasevich, R; Markowich, PA; Shahgholian, H, 2004, p. 1-5Conference paper (Refereed)
    Abstract [en]

    We give a new proof of the fact that the free boundary for the obstacle problem in two dimensions satisfies a natural and sharp inner ball condition.

  • 30.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Sakai, M.
    On the curvature of the free boundary for the obstacle problem in two dimensions2004In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 142, no 02-jan, p. 1-5Article in journal (Refereed)
    Abstract [en]

    We give a new proof of the fact that the free boundary for the obstacle problem in two dimensions satisfies a natural and sharp inner ball condition.

  • 31.
    Gustafsson, Björn
    et al.
    KTH, Superseded Departments, Mathematics.
    Sakai, M.
    Sharp estimates of the curvature of some free boundaries in two dimensions2003In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 28, no 1, p. 123-142Article in journal (Refereed)
    Abstract [en]

    We prove that the Cauchy transform of a positive measure on the interval )-1,1) subset of R in the complex plane maps the exterior of the unit disc onto a domain Omega subset of C which can be written as a union of discs centered on the real axis. This is applied to the obstacle problem, partial balayage, quadrature domains and Hele-Shaw flow moving boundary problems, and we obtain sharp estimates of the curvature of free boundaries appearing in such problems.

  • 32.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sakai, Makoto
    Tokyo Metropolitan University.
    On the curvature of some free boundaries in higher dimensions2012In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 2, p. 247-275Article in journal (Refereed)
    Abstract [en]

    It is known that any subharmonic quadrature domain in two dimensions satisfies a natural inner ball condition, in other words there is a specific upper bound on the curvature of the boundary. This result directly applies to free boundaries appearing in obstacle type problems and in Hele-Shaw flow. In the present paper we make partial progress on the corresponding question in higher dimensions. Specifically, we prove the equivalence between several different ways to formulate the inner ball condition, and we compute the Brouwer degree for a geometrically important mapping related to the Schwarz potential of the boundary. The latter gives in particular a new proof in the two dimensional case.

  • 33.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sebbar, Ahmed
    Critical Points of Green's Function and Geometric Function Theory2012In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 61, no 3, p. 939-1017Article in journal (Refereed)
    Abstract [en]

    We study questions related to critical points of the Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding Mobius invariant differential operators. In the doubly connected case the three Eisenstein series E-2, E-4, E-6 are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these.

  • 34.
    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.

  • 35.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Teodorescu, Razvan
    University of South Florida.
    Vasiliev, Alexander
    Classical and Stochastic Laplacian Growth2014 (ed. 1)Book (Refereed)
  • 36.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tkachev, Vladimir
    Linköpings Universitet.
    On the exponential transform of lemniscates2011In: Notions of Positivity and the Geometry of Polynomials / [ed] P. Bränden, M. Passare, M. Putinar, Basel: Springer, 2011, p. 239-257Chapter in book (Refereed)
    Abstract [en]

    It is known that the exponential transform of a quadrature domain is a rational function for which the denominator has a certain separable form. In the present paper we show that the exponential transform of lemniscate domains in general are not rational functions, of any form. Several examples are given to illustrate the general picture. The main tool used is that of polynomial and meromorphic resultants.

  • 37.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tkachev, Vladimir
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the exponential transform of multi-sheeted algebraic domains2011In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 11, no 2, p. 591-615Article in journal (Refereed)
    Abstract [en]

    We introduce multi-sheeted versions of algebraic domains and quadrature domains, allowing them to be branched covering surfaces over the Riemann sphere. The two classes of domains turn out to be the same, and the main result states that the extended exponential transform of such a domain agrees, apart from some simple factors, with the extended elimination function for a generating pair of functions. In an example we discuss the algebraic curves associated to level curves of the Neumann oval, and determine which of these give rise to multi-sheeted algebraic domains.

  • 38.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tkachev, Vladimir
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On the Jacobian of the Harmonic Moment Map2009Article in journal (Refereed)
    Abstract [en]

    In this paper we represent harmonic moments in the language of transfinite functions, that is projective limits of polynomials in infinitely many variables. We obtain also an explicit formula for the Jacobian of a generalized harmonic moment map.

  • 39.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tkachev, Vladimir
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The Resultant on Compact Riemann Surfaces2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, no 1, p. 313-358Article in journal (Refereed)
    Abstract [en]

    We introduce a notion of the resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.

  • 40.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vasil'ev, A.
    Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics2005In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 84, no 6, p. 551-558Article in journal (Refereed)
    Abstract [en]

    We consider multidimensional weak and strong Hele-Shaw dynamics Omega(t) of an advancing/receding viscous fluid injected/removed through a single finite point into/from a bounded domain Omega(0). A class of weak solutions is shown to preserve local uniqueness in both directions. Then we also consider strong solutions Omega(t), and show that if Omega(0) is starshaped with respect to a small ball centered on the point of injection, then the evolution Omega(t) exists for all time.

  • 41.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vasil'ev, Alexander
    Univ Bergen, Dept Math, N-5020 Bergen, Norway..
    Complex Analysis and Mathematical Physics Preface2010In: Complex Analysis and Operator Theory, ISSN 1661-8254, E-ISSN 1661-8262, Vol. 4, no 3, p. 471-472Article in journal (Other academic)
  • 42.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vasiliev, AlexanderUniversitetet i Bergen.
    Analysis and Mathematical Physics2009Conference proceedings (editor) (Refereed)
  • 43.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vasiliev, Alexander
    Universitetet i Bergen.
    Conformal and Potential Analysis in Hele-Shaw cells2006 (ed. 1)Book (Refereed)
  • 44.
    Roos, Joakim
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gustafsson, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Partial Balayage on Riemannian ManifoldsManuscript (preprint) (Other academic)
    Abstract [en]

    A general theory of partial balayage on Riemannian manifolds is developed, with emphasis on compact manifolds. Partial balayage is an operation of sweeping measures, or charge distributions, to a prescribed density, and it is closely related to (construction of) quadrature domains for subharmonic functions, growth processes such as Laplacian growth and to weighted equilibrium distributions.

    Several examples are given in the paper, as well as some specific results. For instance, it is proved that, in two dimensions, harmonic and geodesic balls are the same if and only if the Gaussian curvature of the manifold is constant.

1 - 44 of 44
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