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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Partial differential equations and Monge–Kantorovich mass transfer by Evans L.C.

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