14 research outputs found
configurations and hyperbolic systems
In this paper we study the rank-one convex hull of a differential inclusion
associated to entropy solutions of a hyperbolic system of conservation laws.
This was introduced in Section 7 of [Kirchheim, M\"uller, \v{S}ver\'ak, 2003]
and many of its properties have already been shown in [Lorent, Peng,
2019]-[Lorent, Peng, 2020]. In particular, in [Lorent, Peng 2020] it is shown
that the differential inclusion does not contain any configurations. Here
we continue that study by showing that the differential inclusion does not
contain configurations.Comment: Author's Accepted Manuscrip
A multi-material transport problem and its convex relaxation via rectifiable -currents
In this paper we study a variant of the branched transportation problem, that
we call multi-material transport problem. This is a transportation problem,
where distinct commodities are transported simultaneously along a network. The
cost of the transportation depends on the network used to move the masses, as
it is common in models studied in branched transportation. The main novelty is
that in our model the cost per unit length of the network does not depend only
on the total flow, but on the actual quantity of each commodity. This allows to
take into account different interactions between the transported goods. We
propose an Eulerian formulation of the discrete problem, describing the flow of
each commodity through every point of the network. We provide minimal
assumptions on the cost, under which existence of solutions can be proved.
Moreover, we prove that, under mild additional assumptions, the problem can be
rephrased as a mass minimization problem in a class of rectifiable currents
with coefficients in a group, allowing to introduce a notion of calibration.
The latter result is new even in the well studied framework of the
"single-material" branched transportation.Comment: Accepted: SIAM J. Math. Ana
A multi-material transport problem with arbitrary marginals
In this paper we study general transportation problems in , in
which different goods are moved simultaneously. The initial and final
positions of the goods are prescribed by measures , on
with values in . When the measures are finite
atomic, a discrete transportation network is a measure on
with values in represented by an oriented graph
in whose edges carry multiplicities in
. The constraint is encoded in the relation . The cost of the discrete transportation is obtained
integrating on a general function
of the multiplicity. When the initial
data are arbitrary (possibly diffuse) measures, the
cost of a transportation network between them is computed by relaxation of the
functional on graphs mentioned above. Our main result establishes the existence
of cost-minimizing transportation networks for arbitrary data
. Furthermore, under additional assumptions on the
cost integrand , we prove the existence of transportation networks
with finite cost and the stability of the minimizers with respect to variations
of the given data. Finally, we provide an explicit integral representation
formula for the cost of rectifiable transportation networks, and we
characterize the costs such that every transportation network with finite cost
is rectifiable.Comment: In V3 we have added an explicit integral representation formula for
the cost of rectifiable transportation networks and characterized the cost
functionals such that every transportation network with finite energy is
rectifiable. The representation formula is proved in the general framework of
-currents with coefficients in group
On the constancy theorem for anisotropic energies through differential inclusions
In this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent papers (De Lellis et al. in Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335; Tione in Minimal graphs and differential inclusions. Commun Part Differ Equ 7:1–33, 2021). In particular, given a polyconvex integrand f, we define a set of matrices Cf that allows us to rewrite the stationarity condition for a graph with multiplicity as a differential inclusion. Then we prove that if f is assumed to be non-negative, then in Cf there is no T′N configuration, thus recovering the main result of De Lellis et al. (Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335) as a corollary. Finally, we show that if the hypothesis of non-negativity is dropped, one can not only find T′N configurations in Cf, but it is also possible to construct via convex integration a very degenerate stationary point with multiplicity
Regularity for graphs with bounded anisotropic mean curvature
We prove that m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in L^p, p > m, are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a C^{1,1} neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a C^{1,1} (resp. C^1) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, forwhich the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs
On singular strictly convex solutions to the Monge-Amp\`ere equation
We show the existence of a strictly convex function
with associated Monge-Amp\`ere measure represented by a function with a.e. whose Hessian has a singular part. This extends the work [13] and
answers an open question of [14,Sec. 6.2(1)]
On a question of D. Serre
In this paper we give a negative answer to the question posed in D. Serre (Ann. Inst. Henri Poincaré C Anal. Non linéaire 35 (2018) 1209–1234, Open Question 2.1) about possible gains of integrability of determinants of divergence-free, non-negative definite matrix-fields. We also analyze the case in which the matrix-field is given by the Hessian of a convex function