Kato square root problem for degenerate elliptic operators on bounded Lipschitz domains

Abstract

Let n≥2, w be a Muckenhoupt A2(Rn) weight, Ω a bounded Lipschitz domain of Rn, and L:=−w−1div(A∇⋅) the degenerate elliptic operator on Ω with the Dirichlet or the Neumann boundary condition. In this article, the authors establish the following weighted Lp estimate for the Kato square root of L:‖L1/2(f)‖Lp(Ω,vw)∼‖∇f‖Lp(Ω,vw) for any f∈W01,p(Ω,vw) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,p(Ω,vw) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, where p is in an interval including 2, v belongs to both some Muckenhoupt weight class and the reverse Hölder class with respect to w, W01,p(Ω,vw) and W1,p(Ω,vw) denote the weighted Sobolev spaces on Ω, and the positive equivalence constants are independent of f. As a corollary, under some additional assumptions on w, via letting v:=w−1, the unweighted L2 estimate for the Kato square root of L that ‖L1/2(f)‖L2(Ω)∼‖∇f‖L2(Ω) for any f∈W01,2(Ω) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,2(Ω) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, are obtained. Moreover, as applications of these unweighted L2 estimates, the unweighted L2 regularity estimates for the weak solutions of the corresponding degenerate parabolic equations in Ω with the Dirichlet or the Neumann boundary condition are also established.