Dirichlet boundary conditions for elliptic operators with unbounded drift

A. Lunardi, G. Metafune, D. Pallara
2005 Proceedings of the American Mathematical Society  
We study the realisation A of the operator A = ∆ − DΦ, D· in L 2 (Ω, µ) with Dirichlet boundary condition, where Ω is a possibly unbounded open set in R N , Φ is a semi-convex function and the measure dµ(x) = exp(−Φ(x)) dx lets A be formally self-adjoint. The main result is that A : D(A) = {u ∈ H 2 (Ω, µ) : DΦ, Du ∈ L 2 (Ω, µ), u = 0 at ∂Ω} is a dissipative self-adjoint operator in L 2 (Ω, µ).
doi:10.1090/s0002-9939-05-08068-8 fatcat:i4qgkmqwozh43m6yolrzbgb2y4