The diamagnetic inequality for the Dirichlet‐to‐Neumann operator

Abstract

Let Ω $\Omega$ be a bounded domain of R d $\mathbb {R}^d$ with Lipschitz boundary Γ $\Gamma$ . We define the Dirichlet-to-Neumann operator N $\mathcal {N}$ on L 2 ( Γ ) $L_2(\Gamma )$ associated with a second-order elliptic operator A = − ∑ k , j = 1 d ∂ k ( c k l ∂ l ) + ∑ k = 1 d ( b k ∂ k − ∂ k ( c k · ) ) + a 0 $A = -\sum _{k,j=1}^d \partial _k (c_{kl} \, \partial _l) + \sum _{k=1}^d (b_k \, \partial _k - \partial _k(c_k \cdot )) + a_0$ . We prove a criterion for invariance of a closed convex set under the action of the semigroup of N ${\cal N}$ . Roughly speaking, it says that if the semigroup generated by − A $-A$ , endowed with Neumann boundary conditions, leaves invariant a closed convex set of L 2 ( Ω ) $L_2(\Omega )$ , then the ‘trace’ of this convex set is invariant for the semigroup of N ${\cal N}$ . 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 ) $L_2(\Gamma )$ .