On the design of a reflector antenna II

By Xu-jia Wang. 2004

146 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/2d9ea630f0c8531ad367230337ccb2ee24cafe44

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In Applied Mathematics

In Numerical Analysis

Citations in the corpus

Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “The regularity results discussed so far — Lipschitz continuity of the potential u, and of Graph(G) ⊂M+ ×M− rather than of the map G(x) = Y (x,Du(x)) itself — required no hypotheses on the probability measures µ− = G#µ +.”
Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna (2013, background): “For this case it has already been checked (see [26, 27, 14]) that the cost function c(x, y) = − log |x − y| satisfies As for x 6= y.”
On the second boundary value problem for Monge-Ampère type equations and optimal transportation (2006, unknown): “where g0 denotes the metric on Sn. See [W1, W2 ,GW, MTW] for more details.”
Hölder regularity of optimal mappings in optimal transportation (2009, background): “For the reflector problem, which is a special optimal transportation problem [ 27 ], the C 1 regularity was obtained by Caffarelli et al. [8].”
Regularity of Potential Functions of the Optimal Transportation Problem (2005, methodology): “In [ 33 ] it was proved that problem (7.4) can be reduced to an optimal transportation problem with cost function”
Continuity, curvature, and the general covariance of optimal transportation (2007, background): “The restriction of the cost function c(x, x̄) = − log |x − x̄| from R × R to the unit sphere M = M̄ = Sn−1 := ∂Bn1(0) arises in conjunction [53], [24] with the reflector antenna problem studied by Caffarelli, Glimm, Guan, Gutierrez, Huang, Kochengin, Marder, Newman, Oliker, Waltmann, Wang, and Wesco”
Interior C 2,α Regularity for Potential Functions in Optimal Transportation (2009, background): “1) arises in optimal transportation [14, 20] and also in reflector and refractor shape design problems [8, 22].”
On the regularity of solutions of optimal transportation problems (2009, background): “Wang [36, 37] (see also Guan and Wang [18] where the results are extended to higher dimension) has shown classical C (and hence C∞) regularity of solutions of the associated Monge-Ampère equation when the densities are smooth.”
Optimal transportation, topology and uniqueness (2010, background): “Adding further restrictions beyond this twist hypothesis allowed Ma, Trudinger, Wang [72,100], and later Loeper [68], to develop a regularity theory for the map f1 : X −→ Y , embracing Delanoë [36], Caffarelli [20,21] and Urbas’ [102] results for the quadratic cost, Gangbo and McCann’s for its restr”