Gradient estimates for higher order elliptic equations on nonsmooth domains

Abstract

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical W m,p , m = 1, 2, . . . , 1 < p < ∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.

Uniform boundary estimates in homogenization of higher-order elliptic systems

“…Theorem 1.2 generalizes the uniform W 1, p estimates for second-order elliptic systems to higher-order elliptic systems. It also extends, in some sense, the W m, p estimate for higher-order elliptic equations (or systems) with non-oscillating coefficients, see e.g., [10,12,13].…”

“…for any 0 < δ < 1/4. Observe that when p > max{d/(m + 1), 1}, L p (D 2 ; R n ) → W −m,q (D 2 ; R n ) for some q > d. Combining the C m−1,λ 0 estimate for higher-order elliptic systems with constant coefficients in C 1 domains (see e.g., [10,12]) and a localization argument (see e.g., the proof of Corollary 5.1), we have…”

“…Proof It is enough to assume 0 < ε < 1/2, as the other case is trivial. Setting [10,12] and a localization argument, we have for any 0…”

Global Gradient Estimates for Nonlinear Elliptic Equations

“…We remark that there are similar technical lemmas in the unweighted case for higher order linear problems, see [7,11,13].…”

“…From a technical point of view, this paper appropriately applies the approach introduced in [10] and later developed in [3,6,7]. Although the main tools are the Hardy-Littlewood maximal function and the Calderón-ZygmundKrylov-Safonov-type decomposition, the general theory of singular integrals employed in [15,22,24] is not used.…”