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1. Hedenmalm, Håkan

An off-diagonal estimate of Bergman kernels2000In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 79, no 2, p. 163-172Article in journal (Refereed)

Abstract [en]

For weights on the unit disk that are logarithmically subharmonic and reproducing for the origin, an estimate from above and below is obtained for the Bergman kernel associated with the weight. In particular, that kernel is zero free, and bounded from above by twice the unweighted Bergman kernel.

The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the Laplacian operator and therefore, the problem of finding a Riemannian metric for a given curvature form may be viewed as a potential theory problem. This problem has, generally speaking, a multitude of solutions. To specify the solution uniquely, we ask that the metric have the mean value property for harmonic functions with respect to some given point. This means that we assume that the surface is simply connected and that it has a smooth boundary. In terms of the so-called metric potential, we are looking for a unique smooth solution to a nonlinear fourth order elliptic partial differential equation with second order Cauchy data given on the boundary. We find a simple condition on the curvature form which ensures that there exists a smooth mean value surface solution. It reads: the curvature form plus half the curvature form for the hyperbolic plane (with the same coordinates) should be less than or equal to 0. The same analysis leads to results on the question of whether the canonical divisors in weighted Bergman spaces over the unit disk have extraneous zeros. Numerical work suggests that the above condition on the curvature form is essentially sharp. Our problem is in spirit analogous to the classical Minkowski problem, where the sphere supplies the chart coordinates via the Gauss map.

Hele-Shaw flow on hyperbolic surfaces2002In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 81, no 3, p. 187-222Article in journal (Refereed)

Abstract [en]

Consider a complete simply connected hyperbolic surface. The classical Hadamard theorem asserts that at each point of the surface, the exponential mapping from the tangent plane to the surface defines a global diffeomorphism. This can be interpreted as a statement relating the metric flow on the tangent plane with that of the surface. We find an analogue of Hadamard's theorem with metric flow replaced by Hele-Shaw flow, which models the injection of (two-dimensional) fluid into the surface. The Hele-Shaw flow domains are characterized implicitly by a mean value property on harmonic functions.

4.

Korvenpaa, Janne

et al.

Aalto Univ, Dept Math & Syst Anal, POB 11100, FI-00076 Aalto, Finland..

Kuusi, Tuomo

Aalto Univ, Dept Math & Syst Anal, POB 11100, FI-00076 Aalto, Finland..

Lindgren, Erik

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

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional p-Laplace type P.V. integral(Rn)vertical bar u(x) - u(y)vertical bar(p-2)(u(x) - u(y))/vertical bar x-y vertical bar(n+sp) dy = 0. Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide. (C) 2017 Elsevier Masson SAS. All rights reserved.