Bartłomiej Siudeja

Maximizing Neumann fundamental tones of triangles

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proven for the harmonic and arithmetic means of the first two nonzero eigenvalues.

Sums of Laplace eigenvalues: Rotations and tight frames in higher dimensions

The sum of the first n ⩾ 1 eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among ellipsoids for the ball, provided the volume and moment of inertia of an auxiliary body are suitably normalized. This result holds for Dirichlet, Robin, and Neumann eigenvalues. Additionally, the cubical torus is shown to be maximal among flat tori. The proof involves euclidean tight frames generated by orbits of the rotation group of the extremal domain. Among general convex domains, the ball is conjectured to maximize sums of Neumann eigenvalues, provided the volume and moment of inertia of the polar dual are suitably normalized.

Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.

Trace estimates for unimodal Lévy processes

We give two-term approximation for the trace of the Dirichlet heat kernel of bounded smooth open set for unimodal Lévy processes satisfying the weak scaling conditions.

The shape of the fundamental sloshing mode in axisymmetric containers

We numerically study positions of high spots (extrema) of the fundamental sloshing mode of a liquid in an axisymmetric tank. Our approach is based on a linear model and reduces the problem to an appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method for making images of an oscillating fluid.