We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in
bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We …

We present some new bounds for the first Robin eigenvalue with a negative boundary
parameter. These include the constant volume problem, where the bounds are based on the …

We give a simple proof of the Faber–Krahn inequality for the first eigenvalue of the p-
Laplace operator with Robin boundary conditions. The techniques introduced allow to work …

The aim of this paper is twofold: First, we characterize an essentially optimal class of
boundary operators Θ which give rise to self-adjoint Laplacians− ΔΘ, Ω in L2 (Ω; dnx) with …

Shape optimization conjectures for the first two eigenvalues of the Robin Laplacian are
developed and supported with new results for rectangular boxes. The square minimizes the …

In this note we prove that the ball is a maximiser for integer order Schatten p-norms of the
Riesz potential operators among all domains of a given measure in R d. In particular, the …

We consider the problem of minimizing the k th eigenvalue of rectangles with unit area and
Dirichlet boundary conditions. This problem corresponds to finding the ellipse centred at the …

The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among
simply-connected planar domains of fixed area when the Robin parameter is scaled by …

We consider the problem of minimising the nth-eigenvalue of the Robin Laplacian in RN.
Although for n= 1, 2 and a positive boundary parameter α it is known that the minimisers do …

One of the principal topics of this paper concerns the realization of self-adjoint operators L Θ,
Ω in L 2 (Ω; dnx) m, m, n∈ ℕ, associated with divergence form elliptic partial differential …