Dirichlet boundary conditions for elliptic operators with unbounded drift

Abstract:

We study the realisation $A$ of the operator $\mathcal {A} = \Delta - \langle D\Phi , D\cdot \rangle$ in $L^2(\Omega , \mu )$ with Dirichlet boundary condition, where $\Omega$ is a possibly unbounded open set in $\mathbb {R}^N$, $\Phi$ is a semi-convex function and the measure $d\mu (x) = \exp (-\Phi (x)) dx$ lets $\mathcal {A}$ be formally self-adjoint. The main result is that $A:D(A)= \{u\in H^2(\Omega , \mu ): \langle D\Phi , Du \rangle \in L^2(\Omega , \mu ), u=0$ at $\partial \Omega \}$ is a dissipative self-adjoint operator in $L^2(\Omega , \mu )$.