Markov semi-groups generated by elliptic operators with divergence-free drift

Semigroups generated by elliptic operators in non-divergence form on C0()

Let ⇢ Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by

Semi-groups of Operators

Symmetry Results for Nonlinear Elliptic Operators with Unbounded Drift

We prove the one-dimensional symmetry of solutions to elliptic equations of the form −div(ea(|∇u|)∇u) = f(u)e, under suitable energy conditions. Our results hold without any restriction on the dimension of the ambient space.

Boundary value problems for elliptic operators with singular drift terms

Let Ω be a Lipschitz domain in R, n ≥ 3, and L = divA∇ − B∇ be a second order elliptic operator in divergence form with real coefficients such that A is a bounded elliptic matrix and the vector field B ∈ Lloc(Ω) is divergence free and satisfies the growth condition dist(X, ∂Ω)|B(X)| ≤ ε1 for ε1 small in a neighbourhood of ∂Ω. For these elliptic operators we will study on the basis of the theory...

Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators

We study quantum dynamical semigroups generated by noncommutative unbounded elliptic operators which can be written as Lindblad type unbounded generators. Under appropriate conditions, we first construct the minimal quantum dynamical semigroups for the generators and then use Chebotarev and Fagnola’s sufficient conditions for conservativity to show that the semigroups are conservative.