Search

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is 8 π k , which is the best upper bound for the k th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For k = 1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point. [ABSTRACT FROM AUTHOR]

In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus γ and the area. The equality in Yang–Yau's estimate is attained for γ = 0 by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for γ = 2 . In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang–Yau's inequality is strict for all genera γ > 2 . Previously this was only known for γ = 1 . In the second part of the paper we apply Chern-Wolfson's notion of harmonic sequence to obtain an upper bound on the total branching order of harmonic maps from surfaces to spheres. Applications of these results to extremal metrics for eigenvalues are discussed. [ABSTRACT FROM AUTHOR]