On the Dirichlet and Neumann Evolution Operators in $\mathbb{R}_{+}$

Abstract

We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators $G_{\mathcal {D}}(t,s)$ and $G_{\mathcal {N}}(t,s)$ associated with a class of nonautonomous elliptic operators (t) with unbounded coefficients defined in I×$\mathbb{R}_{+}$ (where I is a right-halfline or I=ℝ). We also prove the existence and the uniqueness of a tight evolution system of measures $\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}$ associated with $G_{\mathcal {N}}(t,s)$, which turns out to be sub-invariant for $G_{\mathcal {D}}(t,s)$, and we study the asymptotic behaviour of the evolution operators $G_{\mathcal {D}}(t,s)$ and $G_{\mathcal {N}}(t,s)$ in the L ^p-spaces related to the system $\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}$.