Let Ω⊆ℝN be a bounded open set with smooth boundary, and let λ∈ℝ. The Dirichlet-to-Neumann operator Dλ is a closed operator on L2(∂Ω) defined as follows. Given φ∈H1∕2(Ω) solve the Dirichlet problem
A solution exists if λ is not an eigenvalue of -Δ with Dirichlet boundary conditions. If u is smooth enough we define
where ν is the outer unit normal to ∂Ω. Let 0<λ1<λ2<λ3<… be the strictly ordered Dirichlet eigenvalues of -Δ on Ω. It was shown by Arendt and Mazzeo that e-tDλ is positive and irreducible if λ<λ1. The question left open was whether or not the semigroup is positive for any λ>λ1. The aim of this talk is to explore this question by explicitly computing the semigroup for the disc in ℝ2. The example demonstrates some new phenomena: the semigroup e-tDλ can change from not positive to positive between two eigenvalues. This happens for λ∈(λ3,λ4). Moreover, it is possible that e-tDλ is positive for large t, but not for small t. The occurrence of such eventually positive semigroups seems to be new.
A preprint is available.