Asymptotics for Solutions of Elliptic Equations in Double Divergence Form

Abstract

We consider weak solutions of an elliptic equation of the form ∂ i ∂ i (a ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a ij (x) − δ ij ) on the annulus B 2r \\ B r is bounded by a function Ω(r), where Ω2(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L p -mean as r → 0.