Some Remarks on Some Second-Order Elliptic Differential Equations

Abstract

We are concerned with the almost automorphic solutions to the second-order elliptic differential equations of type $\ddot{u}(s)+2B\dot{u}(s)+Au(s)=f(s)(\ast ),$ where $A,B$ are densely defined closed linear operators acting in a Hilbert space $\mathbb{H}$ and $f:\mathbb{R}\mapsto \mathbb{H}$ is a vector-valued almost automorphic function. Using invariant subspaces, it will be shown that under appropriate assumptions; every solution to $(\ast )$ is almost automorphic.

9 pages, first report