On a class of self-adjoint elliptic operators in L2 spaces with respect to invariant measures

AbstractWe consider the operator Au=Δu/2−〈DU,Du〉, where U is a convex real function defined in a convex open set Ω⊂RN and lim|x|→∞U(x)=+∞. Setting μ(dx)=exp(−2U(x))dx, we prove that the realization of A in L2(Ω,μ) with domain {u∈H2(Ω,μ):〈DU,Du〉∈L2(Ω,μ),∂u/∂n=0 at Γ1}, is a self-adjoint dissipative operator. Here Γ1 is the set of points y in the boundary of Ω such that lim supx→yU(x)<+∞. Then we discuss several properties of A and of the measure μ, including Poincaré and log-Sobolev inequalities in H1(Ω,μ)