Sobolev-like Hilbert spaces induced by elliptic operators

Abstract

We investigate properties of function spaces induced by the inner product Sobolev spaces $H^s(\Omega )$ over a bounded Euclidean domain $\Omega$ and by an elliptic differential operator $A$ on $\overline{\Omega }$. The domain and the coefficients of $A$ are of the class $C^{\infty }$. These spaces consist of all distributions $u\in H^s(\Omega )$ such that $Au\in H^{\lambda }(\Omega )$ and are endowed with the corresponding graph norm, with $s,\lambda \in \mathbb{R}$. We prove an interpolation formula for these spaces and discuss their application to elliptic boundary-value problems.

9 pages