The diamagnetic inequality for the Dirichlet-to-Neumann operator

Let $\Omega$ be a bounded domain in R d with Lipschitz boundary $\Gamma$. We define the Dirichlet-to-Neumann operator N on L 2 ($\Gamma$) associated with a second order elliptic operator A = -- d k,j=1 $\partial$ k (c kl $\partial$ l) + d k=1 b k $\partial$ k -- $\partial$ k (c k $\times$) + 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 ($\Omega$), 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 ($\Gamma$).