Uniqueness of second-order elliptic operators with unbounded and degenerate coefficients in L1-spaces

Abstract

Let d ∈ ℕ. Let C = (c_kl)_1≤k,l≤d ∈ W^(1,∞)_loc(ℝ^d, ℝ^(d×d)), W = (w_k)_1≤k≤d ∈ W^(1,∞)_loc(ℝ^d, ℝ^d) and V ∈ L^∞_loc(ℝ^d, ℝ). Consider the formal second-order differential operator Au = -div(C∇u) + W∇u + Vu in L^1(ℝ^d). We show that the closure of (A, C_c^∞(ℝ^d)) is quasi-m-accretive under certain conditions on the coefficients.