Fine Regularity for Elliptic Systems with Discontinuous Ingredients Dian Palagachev and Lubomira Softova

We propose results on interior Morrey, BMO and Hölder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space L. 1. Main Results It is well known that, when dealing with elliptic systems, in contrast to the case of a single second-order elliptic equation, the solely essential boundedness of the principal coefficients is not sufficient to ensure Hölder continuity even of the solution (see [8, Chapter 1]). On the other hand, precise estimates on Hölder’s seminorms of the solution and its lower order derivatives is a matter of great concern in the study of nonlinear elliptic systems. In fact, these bounds imply good mapping properties of certain Carathéodory operators ensuring this way the possibility to apply the powerful tools of the nonlinear analysis and differential calculus. It turns out that “suitable continuity” of the principal coefficients of the system under consideration is sufficient to guarantee good regularity (e.g. Sobolev) of the solutions (see [5, 12]). We deal here with discontinuous coefficients systems for which the discontinuity is expressed in terms of appurtenance to the class of functions with vanishing mean oscillation. Although such systems have been already studied in Sobolev spaces W 2b,p (cf. [3]), our functional framework is that of the Sobolev– Morrey classes W . These possess better embedding properties into Hölder spaces than W 2b,p and as outgrowth of suitable Caccioppoli-type estimates, we give precise characterization of the Morrey, BMO or Hölder regularity of the solution and its derivatives up to order 2b− 1. Let Ω be a domain in R, n ≥ 2, and consider the linear system (1.1) L(x,D)u := ∑ |α|=2b Aα(x)D u(x) = f(x) for the unknown vector-valued function u : Ω → R given by the transpose u(x) = ( u1(x), . . . , um(x) )T , m ≥ 1, f(x) = (f1(x), . . . , fm(x)) , where Aα(x) is the m × mmatrix { a α (x) }m j,k=1 and a α : Ω → R are measurable functions. Hereafter, b ≥ 1 is 1991 Mathematics Subject Classification. Primary 35J45; Secondary 35R05, 35B45, 35B65, 46E35.