Hessian estimates for equations involving p-Laplacian via a fundamental inequality

Abstract

In this paper, we prove a fundamental inequality for the algebraic structure of ΔvΔ∞v: for every v∈C∞,||D2vDv|2−ΔvΔ∞v−12[|D2v|2−(Δv)2]|Dv|2|≤n−22[|D2v|2|Dv|2−|D2vDv|2], where Δ is the Laplacian and Δ∞ is the ∞-Laplacian. Based on this, we prove the following results:(i)For any p-harmonic functions u with p∈(1,2)∪(2,∞), we have|Du|p−γ2Du∈Wloc1,2with γ<min⁡{p+nn−1,3+p−1n−1}. As a by-product, when p∈(1,2)∪(2,3+2n−2), we give a new proof of the known Wloc2,q-regularity of p-harmonic functions for some q>2.(ii)When n≥2 and p∈(1,2)∪(2,3+2n−2), the viscosity solutions to the parabolic normalized p-Laplace equation have the Wloc2,q-regularity in the spatial variable and the Wloc1,q-regularity in the time variable for some q>2. Especially, when n=2 an open question in [18] is completely answered.(iii)When n≥1 and p∈(1,2)∪(2,3), the weak or viscosity solutions to the parabolic p-Laplace equation have the Wloc2,2-regularity in the spatial variable and the Wloc1,2-regularity in the time variable. The range of p here (including p=2 from the classical result) is sharp for the Wloc2,2-regularity.