1. CJM 2010 (vol 62 pp. 808)
|| Extrema of Low Eigenvalues of the Dirichlet-Neumann Laplacian on a Disk|
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet--Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.
Keywords: Laplacian, eigenvalues, Dirichlet-Neumann mixed boundary condition, Zaremba's problem
2. CJM 2009 (vol 62 pp. 109)
||Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues |
Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices
with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$.
All possible inertia values, ranks, and multiple eigenvalues
of $A + B$ are determined. Extension of the results to the sum of $k$ matrices
with $k > 2$ and connections of the results to other subjects such
as algebraic combinatorics are also discussed.
Keywords:complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues