By Evans L.C.

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20) which corresponds to the problem of maximizing Kp [u] := 1 u(f + − f − ) − |Du|p dz. p Rn A maximum principle argument shows that sup |up |, |Dup |, |Dup |p ≤ C < ∞ p for some constant C. (Cf. e. |Du |p−2 a weakly ∗ in L∞ . 19). As noted above, a is the Lagrange multiplier from the constraint |Du∗ | ≤ 1. It turns out furthermore that a in fact “contains” the missing information as to the distance d∗ (x). The recipe is to build s∗ by solving a ﬂow problem involving Du∗ , a, etc. 21) for the time-varying vector ﬁeld b(z, t) := −a(z)Du∗ (z) .

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Brenier, A geometric presentation of the semi–geostrophic equations, preprint. [C1] L. Caﬀarelli, A localization property of viscosity solutions of the Monge–Ampere equaqtion, Annals of Math. 131 (1990), 129–134. [C2] L. Caﬀarelli, Interior W 2,p estimates for solutions of the Monge–Ampere equation, Annals of Math. 131 (1990), 135–150. 55 [C3] L. Caﬀarelli, Some regularity properties of solutions to the Monge–Ampere equation, Comm. in Pure Appl. Math. 44 (1991), 965–969. [C4] L. Caﬀarelli, Allocation maps with general cost functions, in Partial Diﬀerential Equations with Applications (ed.

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