A level set approach for the solution of a state-constrained optimal control problem

Summary. State constrained optimal control problems for linear elliptic partial differential equations are considered. The corresponding first order optimality conditions in primal-dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary (interface) between the active and inactive sets as optimization variable. The new algorithm is based on the level set methodology. The speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization. Encouraging numerical results attained by the new algorithm are reported on.Mathematics Subject Classification (1991): 35R35, 49K20, 49Q10, 65K10Revised version received March 19, 2003     

On the motion of rigid bodies in a viscous incompressible fluid

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.AMS Subject Classification: 49Q10m, 35P15, 49J20.     

The scalar curvature problem on the four dimensional half sphere

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.Received: 2 August 2003, Accepted: 10 May 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J60, 35J20, 58J05K. El Mehdi: elmehdik@ictp.trieste.it     

Linear parabolic equations with singular potentials

By means of maximal regularity techniques we study the solvability of a linear heat equation with a time-dependent potential under minimal regularity assumptions.AMS Subject Classification: 35K10, 35K15, 35K65.     

On the optimization of a conservation law

Consider the Cauchy problem for a conservation law and assume that an integral functional on its solution is defined. In this note we obtain an Euler-Lagrange equation for the stationary points of this functional. An application to the optimal management of traffic flows is considered.Received: 29 April 2002, Accepted: 24 October 2002, Published online: 4 September 2003Mathematics Subject Classification (2000): 35L65, 49K20, 90B20The authors thank Stefano Bianchini for useful discussions.     

A direct uniqueness proof for equations involving the p-Laplace operator

 We provide a simple convexity argument for some known uniqueness theorems. Previous proofs were more technical and had to pay attention to the behaviour of solutions near the boundary. Received: 3 May 2002 Mathematics Subject Classification (2000): 35J20, 35J70, 49R05     

Finite elements for transonic potential flows

A finite element method is used for the computation of entropy solutions to the transonic full potential equation. Physically correct solutions with sharp and correctly placed shocks were obtained. (AMS (MOS) 1980 Mathematics subject classifications: 65N30, 76N15, 35M05, 76H05, 49D10, 35A40, 35L67.)     

Stability remarks to the obstacle problem for p-Laplacian type equations

We extend some convergence and L ¹ stability results for the coincidence set to the p-obstacle problem under natural nondegeneracy conditions and without restrictions on p, . We rely on the local regularity of the solution and, as an application, we show the existence of a solution to the thermal membrane problem, and in a limit nonlocal case also its uniqueness for small data.Received: 2 October 2003, Accepted: 10 May 2004, Published online: 10 December 2004Mathematics Subject Classification (2000): 35J85, 49J40, 35R35, 47J20     

Minimal measures, one-dimensional currents and the Monge–Kantorovich problem

In recent works L.C. Evans has noticed a strong analogy between Mather's theory of minimal measures in Lagrangian dynamic and the theory developed in the last years for the optimalmass transportation (or Monge–Kantorovich) problem. In this paper we start to investigate this analogy by proving that to each minimal measure it is possible to associate, in a natural way, a family of curves on the space of probability measures. These curves are absolutely continuous with respect to the metric structure related to the optimal mass transportation problem. Someminimality properties of such curves are also addressed. Mathematics Subject Classification (2000)37J50, 49Q20, 49Q15     

The infinity Laplacian in infinite dimensions

We study three properties of real-valued functions defined on a Banach space: The absolutely minimizing Lipschitz functions, the viscosity solutions of the infinity Laplacian partial differential equation, and the functions which satisfy comparison with cones. We prove that these notions are equivalent, and we show the existence of such functions. These results are new in the infinite-dimensional case.Received: 14 May 2003, Accepted: 4 August 2003, Published online: 2 April 2004Mathematics Subject Classification (2000): 49L25, 35J60, 54C20     

Sharp-interface limit of the Allen-Cahn action functional in one space dimension

We analyze the sharp-interface limit of the action minimization problem for the stochastically perturbed Allen-Cahn equation in one space dimension. The action is a deterministic functional which is linked to the behavior of the stochastic process in the small noise limit. Previously, heuristic arguments and numerical results have suggested that the limiting action should “count” two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, constructions have been used to derive an upper bound for the minimal action which was proved optimal on the level of scaling. In this paper, we prove that for d = 1, the upper bound achieved by the constructions is in fact sharp. Furthermore, we derive a lower bound for the functional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy measures. The proof relies on an extension of earlier results for the related elliptic problem. Mathematics Subject Classification (2000) 49J45, 35R60, 60F10     

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u‖² _H ¹ _(Ω) ²/‖u‖² _L ² _(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes. Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10     

On a periodic Schrödinger equation with nonlocal superlinear part

We consider the Choquard-Pekar equation and focus on the case of periodic potential V. For a large class of even functions W we show existence and multiplicity of solutions. Essentially the conditions are that 0 is not in the spectrum of the linear part –+V and that W does not change sign. Our results carry over to more general nonlinear terms in arbitrary space dimension N2.Mathematics Subject Classification (2000):35Q55, 35Q40, 35J10, 35J20, 35J60, 46N50, 49J35, 81V70in final form: 14 November 2003     

Improved bounds for minimal feedback vertex sets in tournaments

We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on n nodes has at most 1.5949ⁿ minimal FVS. This significantly improves the previously best upper bound of 1.6667ⁿ by Fomin et al. [STOC 2016] and 1.6740ⁿ by Gaspers and Mnich [J. Graph Theory 72 (1):72–89, 2013]. Our new upper bound almost matches the best‐known lower bound of , due to Gaspers and Mnich. Our proof is algorithmic, and shows that all minimal FVS of tournaments can be enumerated in time .     

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本文考虑了一个其产品保修期内免费小修的退化 生产系统的定期检修策略. 系统的退化过程包括三个状态: 可控制状态, 不可控制状态, 故障状态. 过程呆在可控制状态和不可控制状态的时间假设都服从指数分布. 生产系统在固定的时刻t或发生故障时进行检修, 两者以先发生为准. 本文讨论了使单位产品每周期期望成本最小的最优定期检修时间本文考虑了一个其产品保修期内免费小修的退化生产系统的定期检修策略.系统的退化过程包括三个状态:可控制状态,不可控制状态,故障状态.过程呆在可控制状态和不可控制状态的时间假设都服从指数分布.生产系统在固定的时刻t﹡或发生故障时进行检修,两者以先发生为准.本文讨论了使单位产品每周期期望成本最小的最优定期检修时间t﹡,三种特殊情况显示了最优值t的性质.此外,灵敏性分析和数字实例说明了模型中的参数对最优定期检修策略的影响.     

A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

The system of partial differential equations

Theory of Linear Dissipative Scattering Systems with Discrete Time and with Space States of the Class π k

An arbitrary operator-valued function from the generalized Shur class is realized as the transfer function of a minimal optimal and minimal *-optimal dissipative scattering system with the Pontryagin state space. These results generalize Arov's results for the case of the Hilbert state space. Bibliography: 11 titles.     

Numerical solutions of general-relativistic field equations for rapidly rotating neutron stars

Stationary axial symmetric equilibrium configurations rapidly rotating with uniform angular velocity in the framework of general relativity are considered. Sequences of models are numerically computed by means of a computer code that solves the full Einstein equations exactly. This code employs Neugebauer’s minimal surface formalism, where the field equations are equivalent to two-dimensional minimal surface equations for 4 metric potentials. The calculations are based upon 10 different equations of state. Results of various structures of neutron stars and the rotational effects on stellar structures and properties are reported. Finally some limits to equations of state of neutron stars and the stability for rapidly rotating relativistic neutron stars are discussed.     

Non-Regular Tangential Behaviour of a Monotone Measure

A Radon measure µ on Rⁿ is said to be k-monotone if is a non-decreasing function on (0,) for every x Rⁿ. (If µ is the k-dimensional Hausdorffmeasure restricted to a k-dimensional minimal surface then thisimportant property is expressed by the monotonicity formula.)We give an example of a 1-monotone measure µ in R² withnon-unique and non-conical tangent measures at a point. Furthermore,we show that µ can be the one-dimensional Hausdorff measurerestricted to a closed set A R². 2000 Mathematics Subject Classification49Q05, 49Q20 (primary), 28A75, 53A10 (secondary).