L2-regularity theory of linear strongly elliptic Dirichlet systems of order 2m with minimal regularity in the coefficients

In this article, we consider the following Dirichlet system of order 2m: L(x, del)u = f(x) in Omega, del(k)u = 0 on partial derivativeOmega (k = 0,...,m - 1). Here, Q is a smooth bounded domain in R-n and the differential operator L(x, del) given by (1) satisfies the Legendre-Hadamard condition (4). From the general elliptic theory we know that for sufficiently smooth coefficients A(alphabeta)((m)), B-alphabeta((km)), C-alpha((k)) and for f epsilon H-m+s(Omega,R-n), every weak solution u epsilon H-0(m)(Omega,R-N) is actually in Hm+s(Omega,R-N) and satisfies an a priori estimate of the following form: parallel touparallel to(Mm+s)(Omega,R-N) less than or equal to (C) over cap parallel tofparallel to(H-m+s)(Omega,R-N) + (K) over cap parallel touparallel to(L2)(Omega,R-N). 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, del) 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. Here, we prove the above regularity result under the assumptions (2), (8) on the coefficients and we give an explicit representation formula for the regularity constants and (K) over cap (see (10)).