Order Maximum Principle Research Articles

On the geometry of submanifolds in certain warped products

In this paper, we deal with [Formula: see text]-dimensional submanifolds immersed in a slab of a warped product of the type [Formula: see text]. Under suitable constraints on the warping function [Formula: see text] and assuming that such a submanifold [Formula: see text] is either complete or stochastically complete, we apply some maximum principles in order to show that [Formula: see text] must be contained in a slice of [Formula: see text]. In particular, from our results we guarantee the nonexistence of [Formula: see text]-dimensional closed minimal submanifolds immersed in [Formula: see text]. Furthermore, we construct a nontrivial duo-graph in [Formula: see text] which illustrates the importance of our rigidity results.

Submanifolds immersed in Riemannian spaces endowed with a Killing vector field: Nonexistence and rigidity

Assuming suitable constraints on the warping function ρ and on the mean curvature vector field, we apply the Omori-Yau's generalized maximum principle in order to study the behavior of a support function naturally attached to a complete n-dimensional submanifold Σn immersed in a Killing warped product Mn+p×ρR whose base Mn+p has sectional curvature bounded from below. In the case that Σn is compact, we prove the nonexistence of non-minimal compact mean curvature flow solitons with respect to a non-singular Killing vector field. When Σn has codimension 2, we conclude that there do not exist n-dimensional minimal compact submanifolds immersed in Mn+1×ρR whose base is either Mn+1=Rn+1 or Mn+1=Hn+1. We also get that the only minimal topological 2-spheres immersed in S3×ρR are the totally geodesic ones in a slice S3×{t0}. Moreover, we obtain further results concerning the geometry of complete submanifolds immersed in a Killing warped product, under appropriate restrictions on the Bakry-Émery-Ricci tensor, Ricψ, of these submanifolds with respect to the weight function ψ=ln⁡ρ−2.

Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn equation has been an open problem in recent years. This work provides a positive answer by designing and analyzing a class of up to fourth order maximum principle preserving integrators for the Allen-Cahn equation. First, the second order finite difference discretization is applied to the Allen-Cahn equation in the space direction. The obtained semi-discrete system also preserves the maximum principle and the energy dissipation law. Then the fully discrete numerical scheme is obtained by applying the Lawson transformation and the Runge-Kutta integration in the time direction. We define sufficient conditions for explicit integration factor Runge-Kutta scheme to preserve the maximum principle, namely, the Shu-Osher form of the underlying Runge-Kutta scheme has non-negative coefficients αi,j, nondecreasing abscissas ci and the time step size τ>0 satisfies τ{βi,jαi,j}∈[−4,12]. We prove that the proposed method is convergent with order O(τp+h2) in the discrete L∞ norm. A fast solver is then applied to the discrete system to accelerate numerical computations. Various experiments for 1D, 2D and 3D problems are provided to illustrate the high-order convergence and maximum principle preserving of the proposed algorithms over a long time and verify the theoretical analysis.

A general formulation of time-optimal quantum control and optimality of singular protocols

We present a general theoretical framework for finding the time-optimal unitary evolution of the quantum systems when the Hamiltonian is subject to arbitrary constraints. Quantum brachistochrone (QB) is such a framework based on the variational principle, whose drawback is that it only deals with equality constraints. While inequality constraints can be reduced to equality ones in some situations, they usually cannot, especially when a drift field, an uncontrollable part, is present in the Hamiltonian. We first develop a framework based on Pontryagin’s maximum principle (MP) in order to deal with inequality constraints as well. The new framework contains QB as a special case, and their detailed correspondence is given. Second, we address the problem of singular controls, which satisfy MP trivially so as to cause a trouble in determining the optimal protocol. To overcome this difficulty, we derive an additional necessary condition for a singular protocol to be optimal by applying the generalized Legendre–Clebsch condition. Third, we discuss general relations among the drift, the singular controls, and the inequality constraints. Finally, we demonstrate how our framework and results work in some examples. We also discuss the physical meaning of singular controls.

On the classification of MOTS in the de Sitter space

In this paper, we deal with marginally outer trapped surfaces (MOTS) immersed in the de Sitter space \(\mathbb {S}_1^{n+2}\). In this setting we are able to obtain a Simons formula for the null second fundamental form and under some appropriate constraints on the MOTS, we apply a weak maximum principle in order to guarantee that it must be either a totally geodesic submanifold or isometric to an open piece of an isoparametric submanifold with two distinct principal curvatures one of which is simple. In this last case, supposing that the initial data where the MOTS lying is a totally umbilical spacelike hypersurface of \(\mathbb {S}_1^{n+2}\), we conclude that it must be either isometric to a circular cylinder, a hyperbolic cylinder or a Clifford torus.

Linear Weingarten submanifolds in the hyperbolic space

We use suitable maximum principles in order to obtain characterization results concerning n-dimensional linear Weingarten submanifolds immersed with globally flat normal bundle and parallel normalized mean curvature vector field in the $$(n+p)$$ -dimensional hyperbolic space $${\mathbb {H}}^{n+p}$$ . In particular, when $$p=2$$ , we present complete descriptions of these submanifolds.

Optimal control for the spread of infectious disease: The role of awareness programs by media and antiviral treatment

SummaryThis paper presents a detailed mathematical analysis of the role of awareness programs by media and antiviral treatment on the control of infectious diseases. A nonlinear model is proposed in order to control the infectious diseases, and the model is examined under various scenarios of control measures using variety of mathematical techniques including stability, sensitivity, optimal control, and cost‐effectiveness analysis. Considering the constant control parameters, the basic reproduction number is derived and investigated by the existence and stability of equilibria, then the relative impact of each of the constant control parameters are assessed by calculating the sensitivity index of the basic reproduction number. Under the time‐dependent controls, the necessary conditions for the optimal control of the disease are derived using the Pontryagin's maximum principle in order to determine optimal strategies for controlling the spread of the disease. In order to investigate the most cost‐effective control strategy among all the possible strategies, cost‐effectiveness analysis is carried out using numerical simulations and the result reveals that the strategy combination of 3 control measures (the successful campaign of awareness programs by media, controlling effort that alters infectious cases receiving treatment, and strengthening effort made on awareness campaign programs) is the most cost‐effective control strategy and this indicates that implementation of all the 3 control measures is necessary in order to control the disease outbreak.

Rigidity of complete spacelike hypersurfaces in spatially weighted generalized Robertson–Walker spacetimes

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson–Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f-mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient spacetime. In this setting, we also obtain new Calabi–Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.

Uniqueness of Spacelike Hypersurfaces in a GRW Spacetime via Higher Order Mean Curvatures

We study the problem of uniqueness concerning complete spacelike hypersurfaces immersed in a generalized Robertson–Walker (GRW) spacetime, whose fiber obeys suitable curvature constraints. In this setting, we apply some maximum principles in order to guarantee that such a spacelike hypersurface must be a slice of the ambient space, provided that some of their higher order mean curvatures satisfies appropriated controls. Furthermore, we also establish nonparametric results concerning entire spacelike graphs in GRW spacetimes.

Optimal control approach to dengue reduction and prevention in Cali, Colombia

The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision‐maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide‐treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost‐effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd.

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

Our aim is to apply suitable generalized maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces immersed in a locally symmetric Riemannian manifold, whose sectional curvature is supposed to obey standard constraints. In this setting, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

High Order Maximum Principle Preserving Finite Volume Method for Convection Dominated Problems

In this paper, we investigate the application of the maximum principle preserving (MPP) parametrized flux limiters to the high order finite volume scheme with Runge-Kutta time discretization for solving convection dominated problems. Such flux limiter was originally proposed in [Xu, Math. Comp., 2013] and further developed in [Xiong et. al., J. Comp. Phys., 2013] for finite difference WENO schemes with Runge-Kutta time discretization for convection equations. The main idea is to limit the temporal integrated high order numerical flux toward a first order MPP monotone flux. In this paper, we generalize such flux limiter to high order finite volume methods solving convection-dominated problems, which is easy to implement and introduces little computational overhead. More importantly, for the first time in the finite volume setting, we provide a general proof that the proposed flux limiter maintains high order accuracy of the original WENO scheme for linear advection problems without any additional time step restriction. For general nonlinear convection-dominated problems, we prove that the proposed flux limiter introduces up to $O(\Delta x^3+\Delta t^3)$ modification to the high order temporal integrated flux in the original WENO scheme without extra time step constraint. We also numerically investigate the preservation of up to ninth order accuracy of the proposed flux limiter in a general setting. The advantage of the proposed method is demonstrated through various numerical experiments.

Characterizations of Spacelike Hyperplanes in the Steady State Space via Generalized Maximum Principles

We deal with complete spacelike hypersurfaces immersed in the half of the de Sitter space, which models the so-called steady state space. In this setting, under some appropriated constraints on the geometry of such a spacelike hypersurface, we apply suitable generalized maximum principles in order to guarantee that it must be isometric to the Euclidean space.

Characterizations of linear Weingarten spacelike hypersurfaces in Lorentz space forms

In this article, we deal with complete linear Weingarten spacelike hypersurfaces (that is, complete spacelike hypersurfaces whose mean and scalar curvatures are linearly related) immersed in a Lorentz space form. By assuming that the mean curvature attains its maximum and supposing appropriated restrictions on the norm of the traceless part of the second fundamental form, we apply Hopf's strong maximum principle in order to prove that such a spacelike hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of the ambient space.

LINEAR WEINGARTEN HYPERSURFACES WITH BOUNDED MEAN CURVATURE IN THE HYPERBOLIC SPACE

AbstractWe apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.

New Calabi–Bernstein type results in weighted generalized Robertson–Walker spacetimes

We apply suitable generalized maximum principles in order to obtain new Calabi–Bernstein’s type results concerning complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime. Assuming a natural comparison inequality between the weighted mean curvatures of the hypersurface and those of the slices of the timelike bounded region where the hypersurface is supposed to be contained, we get sufficient conditions which guarantee that such a hypersurface must be a slice. Furthermore, we also treat the case when the ambient spacetime is static.

On the geometry of horospheres

The aim of this paper is to investigate Bernstein-type properties of horospheres of the hyperbolic space \mathbb H^{n+1} . Our approach is based on the use of appropriate generalized maximum principles in order to obtain new characterization results of such horospheres. Furthermore, by supposing a linear dependence between support functions naturally attached to a hypersurface, we also establish a classification theorem concerning horospheres and hyperbolic cylinders of \mathbb H^{n+1} .

Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion

In this work, we study the coupling of a culture of microalgae limited by light and an anaerobic digester in a two-tank bioreactor. The model for the reactor combines a periodic day-night light for the culture of microalgae and a classical chemostat model for the digester. We first prove the existence and attraction of periodic solutions of this problem for a 1 day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain timeframe, the control on the system being the dilution rate (the input flow of microalgae in the digester). We apply Pontryagin's Maximum Principle in order to characterize optimal controls, including the computation of singular controls. We present numerical simulations by direct and indirect methods for different light models and compare the optimal 1-day periodic solution to the optimal strategy over larger timeframes. Finally, we also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks.

High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation

In this paper, we propose the parametrized maximum principle preserving (MPP) flux limiter, originally developed in [37], to the semi-Lagrangian finite difference weighted essentially non-oscillatory scheme for solving the Vlasov equation. The MPP flux limiter is proved to maintain up to fourth order accuracy for the semi-Lagrangian finite difference scheme without any time step restriction. Numerical studies on the Vlasov–Poisson system demonstrate the performance of the proposed method and its ability in preserving the positivity of the probability distribution function while maintaining the high order accuracy.

Complete linear Weingarten hypersurfaces immersed in the hyperbolic space

In this paper, we apply the Hopf's strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $\mathbb H^{n+1}$.