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  • 1. Abkar, A.
    et al.
    Hedenmalm, Håkan
    A Riesz representation formula for super-biharmonic functions2001In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 26, no 2, p. 305-324Article in journal (Refereed)
    Abstract [en]

    Let u be a real-valued function defined on the unit disk D. We call u super-biharmonic provided that u is locally integrable and the bi-laplacian Delta (2)u is a positive distribution on D. In this paper, we shall establish a representation formula for super-biharmonic functions, This formula can be regarded as an analogue of the Poisson-Jensen representation formula for subharmonic functions.

  • 2.
    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.

  • 3.
    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    KTH, Superseded Departments, Mathematics.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.

  • 4.
    Hedenmalm, Håkan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Planar Beurling transform and Grunsky inequalities2008In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 33, no 2, p. 585-596Article in journal (Refereed)
    Abstract [en]

    In recent work with Baranov, it was explained how to view the classical Grunsky inequalities in terms of an operator identity, involving a transferred Beurling operator induced by the conformal mapping. The main property used is the fact that the Beurling operator is unitary on L-2(C). As the Beurling operator is also bounded oil L-p(C) for 1 < p < infinity (with so far unknown norm), all analogous operator identity was found which produces a generalization of the Grunsky inequalities to the L-p setting. Here, we consider weighted Hilbert spaces L-theta(2)(C) with weight, vertical bar z vertical bar(2 theta), for 0 <= theta <= 1, and find that the Beurling operator perturbed by adding a Cauchy-type operator acts unitarily on L-0(2) (C). After transferring to the unit disk D with the conformal mapping, we find a generalization of the Grunsky inequalities ill the setting of the space L-theta(2) (D); this generalization seems to be essentially known, but the formulation is new. As a special case, the generalization of the Grunsky inequalities contains the Prawitz theorem used in a recent paper with Shirnorin. We also mention an application to quasiconformal maps.

  • 5.
    Hedenmalm, Håkan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Beurling operator for the hyperbolic plane2012In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 37, no 1, p. 3-18Article in journal (Refereed)
    Abstract [en]

    We find a Beurling operator for the hyperbolic plane, and obtain an L-2 norm identity for it, as well as two-sided L-p estimates.

  • 6. Kilpelaeinen, Tero
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Zhong, Xiao
    Growth estimates through scaling for quasilinear partial differential equations2007In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 32, no 2, p. 595-599Article in journal (Refereed)
    Abstract [en]

    In this note we use a scaling or blow up argument to obtain estimates to solutions of equations of p-Laplacian type.

  • 7. Mikayelyan, Hayk
    et al.
    Shah Gholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hopf's lemma for a class of singular/degenerate PDE-S2015In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 40, no 1, p. 475-484Article in journal (Refereed)
    Abstract [en]

    This paper concerns Hopf's boundary point lemma, in certain C-1,C-Dini-type domains, for a class of singular/degenerate PDE-s, including p-Laplacian. Using geometric properties of levels sets for harmonic functions in convex rings, we construct sub-solutions to our equations that play the role of a barrier from below. By comparison principle we then conclude Hopf's lemma.

  • 8.
    Shekhar, Atul
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tran, Huy
    Wang, Yilin
    REMARKS ON LOEWNER CHAINS DRIVEN BY FINITE VARIATION FUNCTIONS2019In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 44, p. 311-327Article in journal (Refereed)
    Abstract [en]

    To explore the relation between properties of Loewner chains and operties of their driving functions, we study Loewner chains driven by nctions U of finite total variation. Under a slow point condition, we ow the existence of a simple trace gamma and establish the continuity the map from U to gamma with respect to the uniform topology on gamma d to the total variation topology on U. In the spirit of the work of ng [19] and Lind-Tran [10], we also obtain conditions on the driving nction that ensures the trace to be continuously differentiable.

  • 9.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Nonlocalization of operators of schrödinger type2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 141-147Article in journal (Refereed)
    Abstract [en]

    Localization properties are studied for operators of Schrodinger type.

  • 10.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Spherical harmonics and maximal estimates for the Schrodinger equation2005In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 30, no 2, p. 393-406Article in journal (Refereed)
    Abstract [en]

    Maximal estimates are considered for solutions to an initial value problem for the Schrodinger equation. The initial value function is assumed to be a linear combination of products of radial functions and spherical harmonics. This generalizes the case of radial functions. We also replace the solutions to the Schrodinger equation by more general oscillatory integrals.

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