Sobolev-like Hilbert spaces induced by elliptic operators

TL;DR

This paper studies function spaces generated by elliptic operators and Sobolev spaces, providing interpolation formulas and applications to boundary-value problems, enhancing understanding of elliptic PDEs in smooth domains.

Contribution

It introduces a new class of Sobolev-like spaces induced by elliptic operators and derives interpolation formulas for these spaces.

Findings

Established interpolation formulas for the new function spaces.

Analyzed properties of these spaces in relation to elliptic boundary-value problems.

Discussed applications to elliptic PDE boundary-value problems.

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.