Sharp spectral bounds on starlike domains

Abstract

We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber-Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizes the spectral zeta function and heat trace. The normalization on the domain incorporates volume and a computable geometric factor that measures the deviation of the domain from roundness, in terms of moment of inertia and a support functional introduced by Pólya and Szegő.Additional functionals handled by our method include finite sums and products of eigenvalues. The results hold on convex and starlike domains, and for Dirichlet, Neumann or Robin boundary conditions.

The shape of the fundamental sloshing mode in axisymmetric containers

“…In particular the Dirichlet to Neumann eigenvalue comparisons [36,18,15] was already extended to mixed Steklov eigenvalues [6]. Furthermore, bounds for spectral functionals for Neumann eigenvalues naturally extend to the cylindrical sloshing case (see [35,Section 13]), opening a whole new area of research for general containers. Finally, sloshing problem can be used to tackle, seemingly unrelated, spectral problems for nonlocal fractional Laplacian [7,31,33].…”

Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

“…Motivated by this physical interpretation, and in an attempt to prove a summed version of the Pólya conjecture, the eigenvalue sums of the Laplacian have been studied extensively through Berezin-Li-Yau inequalities [19,35], giving results that are asymptotically sharp as j → ∞. In a different direction, geometrically sharp inequalities for Laplace eigenvalue sums (with fixed index j) were developed on starlike domains by the second and third authors [32,33]. The biLaplacian was treated too [39].…”

Magnetic spectral bounds on starlike plane domains

“…The theorem can be strengthened by replacing the maximum of G 0 and G 1 , which we call G, with certain convex combinations of G 0 and G 1 : see our discussion in the case of zero magnetic field [14,Section 9] . Further improvements can be made by choosing a "good" location for the origin, so as to reduce the values of G 0 and G 1 [14,Section 10].…”

“…Deviation can occur in two ways: an oscillatory boundary would make R ′ large and hence G 0 large, whereas an elongated boundary (such as an eccentric ellipse) would force R 4 to vary more than R 2 and hence would make G 1 large. Calculations are generally required in order to determine which of G 0 or G 1 is larger (see [14,Section 10]).…”