Locality and Domination of Semigroups

Abstract

We characterize all semigroups $(T(t){)}_{t\ge 0}$ on $L^2(\Omega )$ sandwiched between Dirichlet and Neumann ones, i.e.: \begin{equation*}\label{eq:san} e^{tΔ_D}\leq T(t)\leq e^{tΔ_N}\quad,\text{for all }t\geq0 \end{equation*} in the positive operators sense. The proof uses the well-known Beurling-Deny and Lejan formula to drop the locality assumption made usually on the form associated with $(T(t){)}_{t\ge 0}$.

The author was supported, for this work, by "Deutscher Akademischer Austausch Dienst"(German Academic Exchange Service) Grant Number: A/11/97482