Affiliation:
1. Department of Mathematics , Hunan University , Changsha , China
Abstract
Abstract
In this article, we introduce a class of geometric optics measure-the
L
q
Σ
{L}_{q}\hspace{0.33em}\Sigma
-reflector measure which arises from an
L
q
{L}_{q}
extension of the
Σ
\Sigma
-reflector measure. And we ask a Minkowski-type problem for this class of measure, called the
L
q
Σ
{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
L
q
Σ
{L}_{q}\hspace{0.33em}\Sigma
-reflector problem.
Subject
General Mathematics,Statistical and Nonlinear Physics
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