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

Abstract

Abstract. In this paper we study the dimension of some measures which are related to the classical Monge's optimal mass transport problem and are solutions of a scalar shape optimization problem. Moreover in the case of maximal dimension we will study the summability of the associate densities.

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.…”

“…Proving some estimate from below for σ γ could be interesting for the approach of this paper. In fact, assume for example that σ γ has an L ∞ density a γ (see for example [15,17]) and that at a point x one has 0 < a γ (x). Then the lower density of the transport set T (γ ) at x satisfies θ * (T (supp γ ), x) > 0 because a γ (x) = lim Because of the above example, we however cannot expect an estimate from below on σ γ for any solution γ of (1.3), but this may hold for example for an element of O 2 (µ, ν).…”

Optimal Shapes and Masses, and Optimal Transportation Problems

“…The following regularity results have been obtained using formula (40) and a property of the support of optimal plans called ciclical monotonicity (see [96], [97]). …”

“…• The first summability result for µ was obtained in [96], where for 1 < p < +∞ only the L p−ε regularity was proved. The summability improvement has been recently obtained in [97] by means of elliptic PDE estimates.…”

Lack of regularity of the transport density in the Monge problem

“…The L p regularity of the transport density σ is proved successively by many authors (see, for instance, [7,8,9,11,18]). In particular, we have the following…”