Parabolic and Elliptic Systems with VMO Coefficients

Abstract

We consider second order parabolic and elliptic systems with leading coefficients having the property of vanishing mean oscillation (VMO) in the spatial variables. An Lq − Lp theory is established for systems both in divergence and non-divergence form. 1. Introduction. The L p theory of second order parabolic and elliptic equations has been studied extensively by many authors for more than fifty years. It is of particular interest not only because of its various important applications in nonlinear equations, but also due to its subtle links with the theory of stochastic processes. For scalar equations, the solvability theory in L p spaces has been well established; see, for example, [7], [8], [11], [25], [3], [18]-[23], [4] and references therein. For elliptic systems with discontinuous coefficients, there are also quite a few results in the literature. Local L p and Hölder estimates of elliptic systems in non-divergence form with VMO coefficients were obtained in [29]. We also would like to bring attention to an interesting paper [5], in which the authors obtain the W 1