Gradient bounds for solutions of elliptic and parabolic equations

Let L be a second order elliptic operator on R with a constant diffusion matrix and a dissipative (in a weak sense) drift b ∈ L loc with some p > d. We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measure μ satisfying the equation Lμ = 0 and that the closure of L in L1(μ) generates a Markov semigroup {Tt}t≥0 with the resolvent {Gλ}λ>0. We prove that, for any Lipschitzian function f ∈ L 1(μ) and all t, λ > 0, the functions Ttf and Gλf are Lipschitzian and |∇Ttf(x)| ≤ Tt|∇f |(x) and |∇Gλf(x)| ≤ 1 λ Gλ|∇f |(x). An analogous result is proved in the parabolic case. Suppose that for every t ∈ [0, 1], we are given a a strictly positive definite symmetric matrix A(t) = (a(t)) and a measurable vector field x 7→ b(t, x) = (b(t, x), . . . , b(t, x)). Let Lt be the elliptic operator on R d given by Ltu(x) = ∑ i,j≤d a(t, x)∂xi∂xju(x) + ∑ i≤d b(t, x)∂xiu(x). (1) Suppose that A and b satisfy the following hypotheses: (Ha) supt∈[0,1] ( ‖A(t)‖ + ‖A(t)‖ ) < ∞, supt∈[0,1] ‖b(t, · )‖Lp(U) < ∞ for every ball U in R with some p > d, p ≥ 2. (Hb) b is dissipative in the following sense: for every t ∈ [0, 1] and every h ∈ R, there exists a measure zero set Nt,h ⊂ R d such that ( b(t, x+ h)− b(t, x), h ) ≤ 0 for all x ∈ R \Nt,h. (Hc) for every t ∈ [0, 1], there exists a Lyapunov function Vt for Lt, i.e., a nonnegative C-function Vt such that Vt(x) → +∞ and LtVt(x) → −∞ as |x| → ∞. We consider the parabolic equation ∂u ∂t = Ltu, u(0, x) = f(x), (2)