Qingbo Huang

Estimates in the generalized Campanato-John-Nirenberg spaces for fully nonlinear elliptic equations

We establish estimates in BMO and Campanato-John-Nirenberg spaces BMOψ for the second derivatives of solutions to the fully nonlinear elliptic equation F(D2u, x) = f (x).

The near field refractor

Abstract We present an abstract method in the setting of compact metric spaces which is applied to solve a number of problems in geometric optics. In particular, we solve the one source near field refraction problem. That is, we construct surfaces separating two homogeneous media with different refractive indices that refract radiation emanating from the origin into a target domain contained in an n − 1 dimensional hypersurface. The input and output energy are prescribed. This implies the existence of lenses focusing radiation in a prescribed manner.

Harnack inequality for the linearized parabolic Monge-Ampère equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation ut − tr((D2φ(x))−1D2u) = 0 on parabolic sections associated with φ(x), under the assumption that the Monge-Ampère measure generated by φ satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group AT (n) × AT (1), where AT (n) denotes the group of all invertible affine transformations on Rn.