4 problems

Let $\Omega$ be a simply connected Lipschitz domain, and let $D$ be a line segment in the boundary. Consider the first mixed Dirichlet-Neumann eigenvalue problem, with Dirichlet co…

Let $T$ be a triangle with mixed boundary conditions: Neumann conditions on two sides and a Dirichlet condition on the remaining side. Consider the first positive non-trivial eigen…

Let $(\Omega,D,N)$ be a triple where $\Omega$ is simply connected, $D$ is a Euclidean line segment, and … Say that $(\Omega,D,N)$ has no hot spots when each first mixed eigenfuncti…

Let the unit disk carry mixed Dirichlet--Neumann boundary conditions, with the Dirichlet part varying among boundary arrangements; the uniform $2$--partition is the arrangement con…