The diamagnetic inequality for the Dirichlet‐to‐Neumann operator

Abstract

Let (Formula presented.) be a bounded domain of (Formula presented.) with Lipschitz boundary (Formula presented.). We define the Dirichlet-to-Neumann operator (Formula presented.) on (Formula presented.) associated with a second-order elliptic operator (Formula presented.). We prove a criterion for invariance of a closed convex set under the action of the semigroup of (Formula presented.). Roughly speaking, it says that if the semigroup generated by (Formula presented.), endowed with Neumann boundary conditions, leaves invariant a closed convex set of (Formula presented.), then the ‘trace’ of this convex set is invariant for the semigroup of (Formula presented.). 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 (Formula presented.).