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Dafermos: Hyperbolic conservation laws in continuum physics. Springer Verlag, 2000. 49. C. De Lellis: Blow-up of the BV norm in the multidimensional Keyfitz and Kranzer system. Duke Math. , 127 (2004), 313–339. 50. L. S. Gelli & L. Granieri: Minimal measures, onedimensional currents and the Monge-Kantorovich problem. it). 40 L. Ambrosio 51. N. De Pauw: Non unicit´e des solutions born´ees pour un champ de vecteurs BV en dehors d’un hyperplan. R. Math. Sci. Acad. Paris, 337 (2003), 249–252. 52. J.

E. Urbas: Regularity of generalized solutions of Monge-Amp`ere equations, Math. , 197 (1988), 365–393. 82. A. Vasseur: Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. , 160 (2001), 181–193. 83. C. Villani: Topics in mass transportation. Graduate Studies in Mathematics, 58 (2004), American Mathematical Society. 84. C. Villani: Optimal transport: old and new. Lecture Notes of the 2005 SaintFlour Summer school. 85. C. Young: Lectures on the calculus of variations and optimal control theory, Saunders, 1969.

K | → 0 and Area(∂Ωk ) ≤ C for all k. Sets of finite perimeter are defined up to sets of measure zero. We normalize E so that ¯ ∩ Br (x) < |Br (x)| 0< E for all x ∈ E and r > 0 There is a well established theory for such sets. The classical reference is [13]. We will consider a set E ⊂ Rn × [0, +∞) that represents the shape of the drop. We denote (x, z) an arbitrary point with x ∈ Rn and z ∈ [0, +∞). Our energy functional reads Jε (E) = Area(∂E ∩ {z > 0}) − β z=0 x χE dx ε (2) (In the following, we will omit the ε in Jε unless it is necessary to stress it out).

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A Course in Combinatorial Optimization by Schrijver A.


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