Dirichlet boundary conditions for elliptic operators with unbounded drift
Alessandra Lunardi, Giorgio Metafune, Diego Pallara · Proceedings of the American Mathematical Society · 2005
We study the realisation A A of the operator A = Δ − ⟨ D Φ , D ⋅ ⟩ \mathcal {A} = \Delta - \langle D\Phi , D\cdot \rangle in L 2 ( Ω , μ ) L^2(\Omega , \mu ) with Dirichlet boundary condition, where Ω \Omega is a possibly unbounded open set in R N \mathbb {R}^N , Φ \Phi is a semi-convex function and the measure d μ ( x ) = exp ( − Φ ( x ) ) d x d\mu (x) = \exp (-\Phi (x))\,dx lets A \mathcal {A} be formally self-adjoint. The main result is that A : D ( A ) = { u ∈ H 2