Regularity properties for Monge transport density and for solutions of some shape optimization problem

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Sharp summability for Monge Transport density via Interpolation (2004, methodology): “It was proved (see [1,6,8]) that the fact that f ∈ L(1) implies also that σ ∈ L(1).”
The Monge problem for strictly convex norms in Rd (2010, background): “In fact, assume for example that σγ has an L∞ density aγ (see for example [15, 17]) and that at a point x one has 0 < aγ (x).”
Integral Estimates for Transport Densities (2004, methodology): “The paper [4] by De Pascale and Pratelli studied how the integrability properties of f = f − f− affect those of the transport density.”
Existence and stability results in the L 1 theory of optimal transportation (2003, background): “The measure σ := aL, the so-called transport density, plays an important rôle in the theory: its uniqueness and its regularity are discussed, under more general assumptions on μ and ν, in [27], [20], [21].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “4Their proof is somehow intermediate between that of [144] and the one we present here: indeed, approximation by atomic measures is also performed in [209], as here, but on both the source and the target measure, which requires a geometric analysis of the transport rays as in [144].”