On the Lq-reflector problem in ℝn with non-Euclidean norm

Abstract

In this article, we introduce a class of geometric optics measure-the LqΣ{L}_{q}\hspace{0.33em}\Sigma -reflector measure which arises from an Lq{L}_{q} extension of the Σ\Sigma -reflector measure. And we ask a Minkowski-type problem for this class of measure, called the LqΣ{L}_{q}\hspace{0.33em}\Sigma -reflector problem. It is shown that the foundations of such measure have been laid by Caffarelli et al. (Reflector problem in endowed with non-Euclidean norm, Arch. Ration. Mech. Anal. 193 (2009), no. 2, 445–473) (for the Σ\Sigma -reflector measure). Inspired by Alexandrov, we present some variational arguments and existence results of solutions to the LqΣ{L}_{q}\hspace{0.33em}\Sigma -reflector problem.

Keywords

• radial function
• the σ-reflector measure
• the lq σ-reflector measure
• variational formulas
• minkowski-type problems
• 52a27
• 53a07
• 52a41