Workshop: Analysis and Mathematical Physics
Kontakt: mikael [dot] persson_sundqvist [at] math [dot] lth [dot] se
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Speaker: Jan Boman (Stockholm University)
Title: Radon transforms supported in hypersurfaces and a conjecture by Arnold
Abstract: A famous lemma in Newton’s Principia says that the area of a segment of a bounded convex domain in the plane cannot depend algebraically on the parameters of the line that defines the segment. Vassiliev extended Newton’s lemma to bounded convex domains in arbitrary even dimensions. In odd dimensions the volume cut out from an ellipsoid by a hyperplane depends not only algebraically but polynomially on the position of the hyperplane. Arnold conjectured in 1987 that ellipsoids in odd dimensions are the only cases in which the volume function in question is algebraic. The special case when the volume function is assumed to be polynomial has been studied in several papers and was settled very recently. Motivated by a totally different problem I recently tried to construct a compactly supported distribution f≠0 whose Radon transform is supported in the set of tangent planes to the boundary surface ∂D of a bounded convex domain D⊂R^n. However, I found that such distributions can exist only if ∂D is an ellipsoid. This result gives a new proof of the abovementioned special case of Arnold’s conjecture.
Speaker: Jonathan Rohleder (Stockholm University)
Title: Eigenvalue inequalities for Laplace and Schrödinger operators
Abstract: Eigenvalues of elliptic differential operators play a natural role in many classical problems in physics and they have been investigated mathematically in depth. For instance, for the Laplacian on a bounded domain it is well-known that its eigenvalues corresponding to a Neumann boundary condition lie below those that correspond to a Dirichlet condition. In the course of time nontrivial improvements of this observation were found by Pólya, Payne, Levine and Weinberger, Friedlander, and others. In this talk we present extensions of some of their results to further boundary conditions or Schrödinger operators.
Speaker: Corentin Léna (Stockholm University)
Title: Eigenvalue variation under non-regular perturbations
Abstract: I will describe the way eigenvalues of some self-adjoint problems approach those of the Dirichlet Laplacian in a planar domain. We will for instance consider the situation where an additional Dirichlet condition is imposed on a small set or the Dirichlet condition changed to Neumann on a small part of the boundary, and also look at the merging of two magnetic flux lines for an Aharonov-Bohm operator.
These problems are connected and cannot be reduced to an analytic perturbation of the domain. Nevertheless, the variation of the eigenvalues can often be computed by using a suitable capacity-type functional. The talk is based on several joint works with L. Abatangelo, V. Bonnaillie-Noël, V. Felli, L. Hillairet and P. Musolino.