Principal eigenvalue of the fractional Laplacian with a large incompressible drift

Abstract

We study the principal Dirichlet eigenvalue of the operator LA = Δα/2+Ab(x) · ∇, on a bounded C 1,1 regular domain D. Here α ∈ (1,2), Δα/2 is the fractional Laplacian, A ∈ ℝ, and b is a bounded d-dimensional divergence-free vector field in the Sobolev space W 1,2d/(d+α)(D). We prove that the eigenvalue remains bounded, as A→ + ∞, if and only if b has non-trivial first integrals in the domain of the quadratic form of Δα/2 for the Dirichlet condition. © 2013 The Author(s).