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On the generalized \(L_q\)-reflector inverse problem with variational methods

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Abstract

In this paper, we introduce the \(L_q\)-reflector measure as a class of geometric optics measures which is a natural \(L_q\) extension of the reflector measure. The \(L_q\)-reflector measure gives rise to one related Minkowski-type problem, namely the \(L_q\)-reflector inverse problem. It asks for necessary and sufficient conditions on a given measure \(\mu \) in order for it to be the \(L_q\)-reflector measure of a reflector. The main concentration properties of reflectors and measures allow us to establish direct variational proofs. As a result, we solve the existence part of the \(L_q\)-reflector inverse problem for all real numbers q.

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Acknowledgements

I would like to appreciate my supervisor, professor Yong Huang for his guidance and the reviewers for their valuable and insightful comments.

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  1. School of Mathematics, Hunan University, Changsha, 410082, Hunan Province, China

    Jing Wang

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  1. Jing Wang
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Correspondence to Jing Wang.

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Wang, J. On the generalized \(L_q\)-reflector inverse problem with variational methods. Geom Dedicata 216, 48 (2022). https://doi.org/10.1007/s10711-022-00710-w

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