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.).

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.

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.).

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.

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.

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.

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.

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.