A New Approach to Absolute Continuity of Elliptic Measure, with Applications to Non-symmetric Equations

Abstract

In the late 1950s and early 1960s, the work of De Giorgi [DeGi] and Nash [N], and then Moser [Mo], initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain 0, a priori only in a Sobolev space W 1, loc (0), were shown to be Ho lder continuous of some order depending just on ellipticity, and maximum principles and Harnack inequalities were established. The Dirichlet problem for such operators, with continuous data on the boundary, was established in [LSW]. This in turn paved the way for a more systematic and detailed study of the properties of the elliptic measures d|L associated to L=div A{ on a domain 0. The classical properties of existence of non-tangential limits of solutions (Fatou type theorems) and comparison principles appeared in [CFMS], but owed a great deal to the earlier work of Carleson [Ca] and Hunt and Wheeden [H-W] on harmonic functions in Lipschitz domains. All the results mentioned above were carried out for elliptic operators L=div A{ where the matrix A=(aij) has bounded measurable coefficients and is symmetric. However, it turns out that the symmetry of the matrix is not needed to get these results: Morrey [Mor] first observed this in connection with the De Giorgi Nash Moser theory; for the results in [CFMS], this fact has not been formally observed until now. With appropriate reformulation in terms of adjoint operators, and adjoint Green's functions, the results of [CFMS] are valid without the symmetry assumption (see Section 1). doi:10.1006 aima.1999.1899, available online at http: www.idealibrary.com on