$L^2$-regularity theory of linear strongly elliptic Dirichlet systems of order $2m$ with minimal regularity in the coefficients

Abstract

In this article, we consider the following Dirichlet system of order 2m: L(x,∇)u = f(x) in Ω, ∇ku = 0 on δΩ (k = 0,...,m - 1). Here, Ω is a smooth bounded domain in Rn and the differential operator L(x, ∇) given by (1) satisfies the Legendre-Hadamard condition (4). From the general elliptic theory we know that for sufficiently smooth coefficients Aαβ(m), Bαβ(km),Cα(k) and for f ∈ H-m+s(Ω,RN), every weak solution u ∈ H0m(Ω,RN) is actually in Hm+s(Ω,RN) and satisfies an a priori estimate of the following form: ||u||Mm+s(Ω,RN) ≤ Ĉ ||f||H-m+s(Ω,RN) + K||u||L2(Ω,RN). The latter a priori estimate is of particular interest in applications to nonlinear PDEs (see, e.g., [6] and [10]). There the coefficients of L(x, ∇) result from a linearization procedure and consequently they cannot be chosen as smooth as one likes. Therefore, e.g. in [10] (Kato), the author cannot use the famous results stated in [4] (Agmon-Douglis-Nirenberg) but refers to [14] (Milani) instead.