Elliptic problems on ?N with unbounded coefficients in classical Sobolev spaces

Abstract

We discuss the existence and uniqueness in H1(ℝN) and the H2(ℝN) regularity of the solutions of Au=f when f ∈ L2(ℝN) and A is a second-order linear elliptic operator whose first and zeroth order coefficients may be unbounded at infinity. We also investigate whether −A generates a C0 or analytic semigroup on L2. The approach in this nonweighted setting is based on a new and general method. The idea consists in embedding A into a suitable one-parameter family of operators (As)s∈ ℝ with A0=A. The properties of As when s≠0 make it possible to prove that the boundary integrals arising from simple integration by parts over balls with increasing radius tend to 0 at infinity. This provides the needed estimates for uniqueness and regularity.