A Dirichlet eigenvalue of a domain 
is an eigenvalue of the positive Laplacian
operator 
with homogeneous Dirichlet boundary conditions.
For a bounded open domain, its weak realization on 
has a discrete positive spectrum 
, repeated according to multiplicity
(Davies 1995). The first eigenvalue is given by the Rayleigh
quotient
 |
It is simple when 
is connected.
Chen et al. (2026) proved that, in every dimension 
, smooth strictly star-shaped
bounded domains can realize any prescribed finite sequence 
of multiplicities of their first distinct Dirichlet
eigenvalues. The eigenvalue values themselves are not prescribed. For bounded convex domains in 
, 
, they proved instead that the second Dirichlet eigenvalue
has multiplicity at most 
, with balls attaining the bound.
