Properties of the solutions of the linearized Monge-Ampère equation

Abstract

Let Φ: R n → R be a function strictly convex and smooth, and μ = det D 2 Φ is the Monge-Ampere generated by Φ. Given x ∈ R n and t > 0, let' S ( x, t ) = { y ∈ R n : Φ( y ) < Φ( x ) + ∇Φ( x ) · ( y - x ) + t }. The purpose of this paper is to study the properties of the solutions of the linearized Monge-Ampère equation given by a ij ( x ) D ij u = 0 where the coefficients a ij ( x ) are the cofactors of the matrix D 2 Φ( x ). It is assumed that μ satisfies a doubling condition on the sets S ( x, t ) and a uniform continuity condition at every scale with respect to Lebesgue measure. We establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant Harnack's inequality on the sections S ( x, t ). All the estimates are independent of the regularity of Φ and depend only on the constants in the hypotheses made on the measure μ.