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 + V u

in L₁(R^d). We show that the closure of (A,C^∞_c (R^d)) is quasi-m-accretive under certain conditions on the coefficients.