L_P-Theory For Higher Order Quasilinear Elliptic Equations With Discontinuous Coefficients

ABSTRACT:In this paper, we prove Lp-regularity theory of the m-th deriva-tives of weak solutions to higher order quasilinear divergence form elliptic equa-tions with discontinuous coefficients defined in a Lipschitz bounded domains. More precisely, we consider the Dirichlet boundary problem to the following higher or-der quasilinear divergence form elliptic equations defined in a bounded domains Ω(?)Rd with Lipschitz continuous boundary (?)Ω: where:a main assumption to the leading coefficients Aαβ(x, u) is allowed Vanish-ing Mean Oscillation dependence on the variable x in small balls and continuous dependence on the variable u∈R with respect to the fixed x(?)Ω. Moreover, they are bounded measurable matrix and satisfy a uniformly ellipticity, that is, there exists a constant μ∈(0,1] such that Here, we denote higher order quasilinear divergence form elliptic operator as fol-lows: and the lower order nonlinear term b(x, u, Du,..., Dmu) satisfies the following controlled growth conditions:|b(x,u,Du,...,Dmu)|≤M(|Dmu|λ1+|u|λ2)+g(x), where λ1=2(1-1/γ), λ2=γ-1and γ=2d/d-2m.This is a extension on higher order quasilinear elliptic equations of Dong’s works on second order quasilinear elliptic equations with controlled growth con-ditions [6] and higher order linear elliptic setting [7]. we study the Lp-regularity theory of the m-th derivatives of weak solutions to higher order quasilinear di-vergence form elliptic equations of discontinuous coefficients with the controlled growth conditions.The main conclusion of this paper is that we will prove that Lp-estimates of the m-th derivatives of its weak solutions in sobolev spaces Wm,2(Ω) for (?)g(x)∈Lp(Ω) with p>d/m. Generally speaking, the higher order numbers of partial differential equations, the more complex the problem. Especially for all kinds of estimates of higher order elliptic equations. The main difficulty here is that the usual methods on the sec-ond order elliptic equations such as maximal principles and De Giorgi-Moser-Nash iterating approach are not suitable to the setting of higher order elliptic equations, which increased the difficulty of the various estimates. Thus, our main technique is to use the perturbation argument. More precisely, we first establish integrability improve for higher order quasilinear divergence form elliptic equations. Then we use the Lp-estimates of higher order linear elliptic equations. Finally we improve the Lp-integrability of the m-th derivatives of its weak solutions by so-called boot-strap argument[31](D. Palagachev, J.M.A.A,2009). As a consequence, we derive Holder continuity of the weak solutions with the exact Holder exponent by Sobolev embedding theorem.