Integral Estimates for Transport Densities

On a Differential Model for Growing Sandpiles with Non-Regular Sources

“…This implies that u T x ≥ u f x , where u f x is defined in (11). So, in view of Proposition 2.2(ii) and assumption (13), to obtain the conclusion it suffices to note that u f x coincides with d.…”

Free boundaries in optimal transport and Monge-Ampère obstacle problems

“…Kellerer [52], Rachev and Rüschendorf [67], or Villani [80], while unique characterizations of optimal maps for other costs have been investigated by Ahmad [3], Ambrosio and Rigot [6], Caffarelli [18], Gangbo and McCann [41], [42], [43], Gangbo andŚwiȩch [44], McCann [60], [61], Plakhov [66], Rüschendorf and Uckelmann [10], Uckelmann [78] in various geometries. Monge's cost c.x; y/ D jx yj in particular has attracted recent attention from Ambrosio [7] and Pratelli [9], Ambrosio, Kircheim and Pratelli [5], Caffarelli,Feldman and McCann [20], DePascale, Evans and Pratelli [30], Feldman and McCann [37], [36], and Trudinger and Wang [76] following the work of Evans and Gangbo [34] and Sudakov [75]. Although the free boundary problem we pose has not been much studied, other transportation problems in which one measure is fixed and the second is selected by a variational principle have been examined extensively in the context of dynamical problems since the work of Otto [65] and Jordan, Kinderlehrer and Otto [50], see also e.g.…”

The Monge problem for strictly convex norms in $\mathbb{R}^d$

“…A detailed discussion of the properties of such measures is beyond the scope of this paper. The transport density plays a crucial role in the proof of existence given in [17], and good estimates from above are available for σ γ [1,15,14,16]. Proving some estimate from below for σ γ could be interesting for the approach of this paper.…”