Stanford Department of Mathematics Analysis & PDE seminar Dirichlet problem for elliptic operators with rough coefficients and L-harmonic measure

Sharp estimates for the solutions to elliptic PDEs in L∞ in terms of the corresponding norm of the boundary data follow directly from the maximum principle. It holds on arbitrary domains for all (real) second order divergence form elliptic operators −divA∇. The wellposedness of boundary problems in L, p < ∞, is a far more intricate and challenging question, even in a half-space. In particular, it is known that some smoothness of A in t, the transversal direction to the boundary, is needed. In the present work we establish the well-posedness in L of the Dirichlet problem for all divergence form elliptic equations with real (possibly non-symmetric) coefficients independent on the transversal direction to the boundary. Equivalently, we show that for all such operators the L-harmonic measure is quantifiably absolutely continuous with respect to the Lebesgue measure. The lack of smoothness and lack of symmetry in the coefficients defy most of the previously known methods. We introduce a new strategy and use the celebrated Kato problem estimate, adapted Hodge decomposition, square function/non-tangential maximal function bounds, epsilon-approximability and A∞ criteria for harmonic measure, among other tools. This is joint work with S. Hofmann, C. Kenig, and J. Pipher. Friday, September 26th, 1:15pm, Room 380-W http://math.stanford.edu/~andras/PDE/PDE.html