Abstract
We prove local \(C^{1,\alpha }\) estimates of solutions to the parallel refractor and reflector problems under local assumptions on the target set \(\Sigma \), and no assumptions are made on the smoothness of the densities.
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Notes
We have in this case, \(\delta =\dfrac{-\kappa +\sqrt{1+(1-\kappa ^2)|v|^2}}{\sqrt{1+|v|^2}}\).
The assumption \(s_{Z}\in C^2\) is need to formulate (3.2) and used to prove that (3.2) and (3.2) are equivalent. However, to formulate (3.2) and for the set up in Sect. 2.5 it is enough to assume that \(s_{Z}\) is Lipschitz. Also Lemmas 5.1 and 5.2 hold true for \(s_{Z}\) Lipschitz because only (3.2) is used.
This condition is implied by the visibility condition in Lemma 2.1 because \(Y\in \Sigma \) and some \(X_0\in C_\Omega \) the line joining \(Y\) and \(X_0\) is contained in \(T_Y\), then there is ball \(B\) centered at \(X_0\) with \(B\subset C_\Omega \). By the visibility condition the convex hull \(\mathcal C\) of \(Y\) and \(B\) intersects \(\Sigma \) only at \(Y\). But then the line joining \(Y\) and \(X_0\) is contained in \(\mathcal C\) and \(T_Y\). Therefore \(\Sigma \) is not differentiable at \(Y\).
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Acknowledgments
It is a pleasure to thank Neil Trudinger and Philippe Delanoë for useful comments and suggestions. We also like to thank the anonymous referee for suggestions and comments that helped to improve the presentation.
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Communicated by N. Trudinger.
The first author was partially supported by NSF grant DMS–1201401.
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Gutiérrez, C.E., Tournier, F. Regularity for the near field parallel refractor and reflector problems. Calc. Var. 54, 917–949 (2015). https://doi.org/10.1007/s00526-014-0811-0
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DOI: https://doi.org/10.1007/s00526-014-0811-0