117 research outputs found
Evolution models for mass transportation problems
We present a survey on several mass transportation problems, in which a given
mass dynamically moves from an initial configuration to a final one. The
approach we consider is the one introduced by Benamou and Brenier in [5], where
a suitable cost functional , depending on the density and on
the velocity (which fulfill the continuity equation), has to be minimized.
Acting on the functional various forms of mass transportation problems can
be modeled, as for instance those presenting congestion effects, occurring in
traffic simulations and in crowd motions, or concentration effects, which give
rise to branched structures.Comment: 16 pages, 14 figures; Milan J. Math., (2012
Improved energy bounds for Schrödinger operators
Given a potential and the associated Schrödinger operator -Δ+, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example or enjoys suitable summability properties, the problem has a positive answer. In this paper we show that the corresponding isoperimetric-like inequalities can be improved by means of quantitative stability estimates
The location of the hot spot in a grounded convex conductor
Abstract. We investigate the location of the (unique) hot spot in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the for-mer is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Ampère equations; the latter relies on Alexandrov’s re-flection principle. We then show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computa-tions. 1
A Benamou-Brenier approach to branched transport
The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length covered by a mass is proportional to with . The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks\dots Several models have been employed in the literature to present this transport problem, and the present paper looks at a dynamical one, similar to the celebrated Benamou-Brenier formulation of Kantorovitch optimal transport. The movement is represented by a path of probabilities, connecting an initial state to a final state , satisfying the continuity equation \partial_t\rho+\dive_xq=0 together with a velocity field (with being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures:
Global compactness results for nonlocal problems
International audienceWe obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional Laplacian operator and nonlinearities at critical growth
The heart of a convex body
We investigate some basic properties of the {\it heart}
of a convex set It is a subset of
whose definition is based on mirror reflections of euclidean
space, and is a non-local object. The main motivation of our interest for
is that this gives an estimate of the location of the
hot spot in a convex heat conductor with boundary temperature grounded at zero.
Here, we investigate on the relation between and the
mirror symmetries of we show that
contains many (geometrically and phisically) relevant points of
we prove a simple geometrical lower estimate for the diameter of
we also prove an upper estimate for the area of
when is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic
and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6
Congested traffic equilibria and degenerate anisotropic PDEs
Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in Baillon and Carlier (Netw. Heterogenous Media 7: 219--241, 2012). Here, we go one step further by showing that these problems can be reformulated in terms of the minimization of an integral functional over a set of vector fields with prescribed divergence. We prove a Sobolev regularity result for their minimizers despite the fact that the Euler-Lagrange equation of the dual is highly degenerate and anisotropic. This somehow extends the analysis of Brasco et al. (J. Math. Pures Appl. 93: 652--671, 2010) to the anisotropic case
Fractional De Giorgi classes and applications to nonlocal regularity theory
We present some recent results obtained by the author on the regularity of
solutions to nonlocal variational problems. In particular, we review the notion
of fractional De Giorgi class, explain its role in nonlocal regularity theory,
and propose some open questions in the subject.Comment: Short note based on a talk given by the author at a conference held
in Bari on May 29-30, 2017, as part of the INdAM intensive period
"Contemporary research in elliptic PDEs and related topics
Developing realistic exposure scenarios for environmental risk rssessment of pesticides in Brazil: surface water
Standard scenarios increase the consistency of the regulatory evaluation process by minimizing the subjective influence when performing the pesticide environmental concentration - PEC calculation, also make interpretation much easier, and enable the adoption of a consistent scientific process for a Tier 1 evaluation (FOCUS, 2000). The Brazilian Institute of the Environment and Renewable Natural Resources (IBAMA) adopts the United States Environmental Protection Agency (USEPA) methodology (IBAMA, 2016), using the ?standard farm pond? scenario (Parker et al., 1995) for surface water in the Tier 1 evaluation, and, therefore, it may lead to the registration of pesticides without considering Brazilian agricultural scenarios in environmental risk assessment - ERA
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