L p -estimates for the solutions of second order elliptic equations

Abstract

We investigate the homogeneous Dirichlet problem in H2,p for a second order elliptic partial differential equation in nondivergence form Lu=f in the case in which the leading coefficients of L belong to H1,n(Ω), Ω ⊂ Rn. We prove that if p belongs to a suitable neighbourhood of 2, then the above problem, has a unique solution u satisfying ∥D2u∥p⩽ C∥f∥p; furthermore, if f e Hk,p, k=1,2, ..., and the coefficients of L satisfy some natural conditions, then the solution satisfies\(\left\| u \right\|_{H^{k + 2,p} } \leqslant C\left\| f \right\|_{H^{k,p} }\).