ELLIPTIC ESTIMATES INDEPENDENT OF DOMAIN EXPANSION

In this paper, we consider elliptic estimates for a system with smooth variable coefficients on a domain \({\Omega \subset \mathbb{R}^n,\, n \ge 2}\) containing the origin. We first show the invariance of the estimates under a domain expansion defined by the scale that \({y = Rx,\, x,\,y \in \mathbb{R}^n}\) with parameter R > 1, provided that the coefficients are in a homogeneous Sobolev space. Then we apply these invariant estimates to the global existence of unique strong solutions to a parabolic system defined on an unbounded domain.