Light reflection is nonlinear optimization

In this paper, we show that the near field reflector problem is a nonlinear optimization problem. From the corresponding functional and constraint function, we derive the Monge-Amp\`ere type equation for such a problem.Comment: to appear in Calc. Var. and PDE

An obstacle problem for a class of Monge-Amp\`ere type functionals

In this paper we study an obstacle problem for Monge-Amp\`ere type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation

A Schauder estimate for stochastic PDEs

Considering stochastic partial differential equations of parabolic type with random coefficients in vector-valued H\"older spaces, we obtain a sharp Schauder estimate. As an application, the existence and uniqueness of solution to the Cauchy problem is also proved.Comment: This is an abridged version. A full version is submitted separatel

A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions

In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic equations arising from a Yamabe problem with boundary and reflector problems