Geodesic
Smooth solutions of the eikonal equation and the~behaviour of local minima of the distance function
Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1234-1258

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We study smooth solutions of the eikonal equation. To do this, we investigate the problem of geometric-topological properties of the singularities of the distance function and the regular set. We establish a connection between the caustic and domains where the number of local minima of the distance function is constant. We pose a number of problems about reflecting surfaces bringing light to a single point (a focus) and introduce the notions of generalized ellipsoids and paraboloids.
Keywords: singular sets, regular sets, solarity points, caustic.
Mots-clés : eikonal equation 
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     author = {I. G. Tsar'kov},
     title = {Smooth solutions of the eikonal equation and the~behaviour of local minima of the distance function},
     journal = {Izvestiya. Mathematics },
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     url = {http://geodesic.mathdoc.fr/item/IM2_2019_83_6_a4/}
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I. G. Tsar'kov. Smooth solutions of the eikonal equation and the~behaviour of local minima of the distance function. Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1234-1258. http://geodesic.mathdoc.fr/item/IM2_2019_83_6_a4/