On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain

Simon Raulot (1) , Alessandro Savo (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 MeMoMat - Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $\Omega$ in Euclidean space in terms of the isoperimetric ratio ${\rm Vol}(\bd\Omega)/{\rm Vol}(\Omega)$.

Mots clés: Manifolds with boundary; Differential forms; Dirichlet-to-Neumann operator; Eigenvalues

Domaines: Géométrie différentielle [math.DG]

Simon Raulot, Alessandro Savo. On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain. Journal of Geometry and Physics, 2014, 77, pp.1-12. ⟨10.1016/j.geomphys.2013.11.002⟩. ⟨hal-00671047⟩