1,720,997 research outputs found
K-mean convex and K-outward minimizing sets
No full textWe consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated) nonlocal perimeters of the discrete evolutions to the nonlocal perimeter of the limit flow
Boundary value problems for Choquard equations
Full text linkWe consider the following nonlinear Choquard equation −Δu+Vu=(Iα∗|u|p)|u|p−2uinΩ⊂RN, where N≥2, p∈(1,+∞), V(x) is a continuous radial function such that infx∈ΩV>0 and Iα(x) is the Riesz potential of order α∈(0,N). Assuming Neumann or Dirichlet boundary conditions, we prove existence of a positive radial solution to the corresponding boundary value problem when Ω is an annulus, or an exterior domain of the form RN∖Br(0) ̄. We also provide a nonexistence result: if p≥[Formula presented] the corresponding Dirichlet problem has no nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting α→0+ we recover existence results for the corresponding local problem with power-type nonlinearity
Minimal periodic foams with fixed inradius
Full text linkIn this note, we show existence and regularity of periodic tilings of the Euclidean space into equal cells containing a ball of fixed radius, which minimize either the classical or the fractional perimeter. We also discuss some qualitative properties of minimizers in dimensions 3 and 4
Nonlocal minimal clusters in the plane
Full text linkWe prove existence of partitions of an open set Ω with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter s is sufficiently close to 1, the only singular minimal cone, that is, the only minimal partition invariant by dilations and with a singular point, is given by three half-lines meeting at 120 degrees. In the case of a weighted sum of fractional perimeters, we show that there exists a unique minimal cone with three phases
Ergodic Mean-Field Games with aggregation of Choquard-type
Full text linkWe consider second-order ergodic Mean-Field Games systems in the whole space RN with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. These MFG systems describe Nash equilibria of games with a large population of indistinguishable rational players attracted toward regions where the population is highly distributed. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for the existence and non existence of classical solutions to the MFG system. By means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical regime. On the other hand, using a fixed point argument, we show existence of classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be +∞
Minimal Periodic Foams with Equal Cells
No full textWe show existence of periodic foams with equal cells in R-n minimizing an anisotropic perimeter
Spontaneous volunteerism in disasters, managerial inputs and policy implications from Italian case studies
Full text linkThe landscape of the emergency response system is rapidly changing, due also to an increasing frequency of natural and man-made disasters. Nowadays, the involvement of citizens into the whole emergency management is inevitable and indispensable, from preparedness to recovery operations. Drawing on international disaster management literature, this research analyzes and compares the management of spontaneous volunteers during six emergency events that recently occurred in Italy. The research's aims are to present current tendencies of the Italian emergency volunteerism, to provide operational recommendations for organizations that may be in charge of managing unaffiliated volunteers and to propose some hints for a reform process of the Italian Civil Protection System towards a recognition of the spontaneous volunteerism in emergency. Starting from the research's findings, the study underlines the latest trends in the field of emergency operations, at a local and international level, and their policy implications, within the Italian context
One-dimensional multi-agent optimal control with aggregation and distance constraints: Qualitative properties and mean-field limit
Full text linkIn this paper we consider an optimal control problem for a large population of interacting agents with deterministic dynamics, aggregating potential and constraints on reciprocal distances, in dimension 1. We study existence and qualitative properties of periodic in time optimal trajectories of the finite agents optimal control problem, with particular interest on the compactness of the solutions' support and on the saturation of the distance constraint. Moreover, we prove, through a Γ-convergence result, the consistency of the mean-field optimal control problemwith density constraintswith the corresponding underlying finite agent one and we deduce some qualitative results for the time periodic equilibria of the limit problem
Graphical translators for anisotropic and crystalline mean curvature flow
No full textIn this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows to find explicitly the translators and describe their main features. (C) 2022 Elsevier Inc. All rights reserved
Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control
Full text linkFor a class of Bellman equations in bounded domains we prove that sub-and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with ergodic control with state constraints. In this context, our condition on the behavior of the operator near the boundary ensures some invariance property of the domain for the associated controlled diffusion process
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