On the L q -reflector problem in ℝ n with non-Euclidean norm

Author:

Wang Jing1

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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

Reference30 articles.

1. A. Aleksandrov, On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sb.(NS) 3 (1938), 27–46.

2. A. Aleksandrov, On the surface area measure of convex bodies, Mat. Sb.(NS) 6 (1939), 167–174.

3. L. Caffarelli, C. E. Gutiérrez and Q. Huang, On the regularity of reflector antennas, Ann. Math. 167 (2008), no. 1, 299–323.

4. L. Caffarelli, A. Luis, and Q. Huang, Reflector problem in Rn endowed with non-Euclidean norm, Arch. Ration. Mech. Anal. 193 (2009), no. 2, 445–473.

5. L. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics, J. Math. Pures Appl. 9 154 (2008), no. 1, 39–49.

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