The diamagnetic inequality for the Dirichlet-to-Neumann operator

2020/04/21 by A. F. M. ter Elst, ter Elst, . A. F. M., El Maati Ouhabaz +1 · 1 citation
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.2004.09782

openalex publication_date 2020/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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).

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