The diamagnetic inequality for the Dirichlet-to-Neumann operator

. A.F.M. ter Elst (1) , El Maati Ouhabaz (2)

1 University of Auckland [Auckland]
2 IMB - Institut de Mathématiques de Bordeaux

Let Ω be a bounded domain in R d with Lipschitz boundary Γ. We define the Dirichlet-to-Neumann operator N on L 2 (Γ) associated with a second order elliptic operator A = − d k,j=1 ∂ k (c kl ∂ l) + d k=1 b k ∂ k − ∂ k (c k ·) + a 0. We prove a criterion for invariance of a closed convex set under the action of the semigroup of N. Roughly speaking, it says that if the semigroup generated by −A, endowed with Neumann boundary conditions, leaves invariant a closed convex set of L 2 (Ω), then the 'trace' of this convex set is invariant for the semigroup of N. We use this invariance to prove a criterion for the domination of semigroups of two Dirichlet-to-Neumann operators. We apply this criterion to prove the diamagnetic inequality for such operators on L 2 (Γ).

Keywords: Dirichlet-to-Neumann operator; invariance of convex sets; domination of semi- groups; diamagnetic inequality Home institutions:

Domains: Analysis of PDEs [math.AP]; Functional Analysis [math.FA]

. A.F.M. ter Elst, El Maati Ouhabaz. The diamagnetic inequality for the Dirichlet-to-Neumann operator. 2020. ⟨hal-02547609⟩