Quasilinear degenerate evolution equations in Banach spaces

The quasilinear degenerate evolution equation of parabolic type 0< t T considered in a Banach space X is written, putting Mv = u, in the from 0< t T, where A(u)=L(u)M^–1 are multivalued linear operators in X for u K, K being a bounded ball ||u||_Z<R in another Banach space Z continuously embedded in X. Existence and uniqueness of the local solution for the related Cauchy problem are given. The results are applied to quasilinear elliptic-parabolic equations and systems.     

Maximal regularity for evolution equations in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem enjoys maximal regularity in weighted L _p -spaces with weights , where , if and only if it has the property of maximal L _p -regularity. Moreover, it is also shown that the derivation operator admits an -calculus in weighted L _p -spaces. Received: 26 February 2003     

Propriétés arithmétiques d'une famille de surfaces K3

We study a family of K3 surfaces which have a big automorphism group. We begin with generalisations of Silverman's results: construction of canonical heights, density of rational points in one orbit,... We continue the study in estimating the density of rational points on the orbiting rational curves; this estimate is compatible with Batyrev–Manin conjecture. Moreover we settle, under more geometric hypothesis, the number of rational points of such surfaces of bounded height.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 1: Problems with No State Constraints

This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints, we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive Pontryagin's principles for problems with pointwise state constraints. Accepted 12 July 2001. Online publication 21 December 2001.     

K3 Surfaces with Nine Cusps

By a K3-surface with nine cusps I mean a surface with nine isolated double points A₂, but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown that such a surface admits a cyclic triple cover branched precisely over the cusps. This parallels the theorem of Nikulin that a K3-surface with 16 nodes is a Kummer quotient of a complex torus.     

On the Kodaira Dimension of the Moduli Space of K3 Surfaces II

We show that the Kodaira dimension of the moduli space of polarized K3 surfaces of degree 2n in non negative if n = 42, 43, 51, 53, 55, 57, 59, 61, 66, 67, 69, 74, 83, 85, 105, 119 or 133. We use an automorphic form associated with the fake monster Lie algebra constructed by Borcherds.     

Datalogy — The copenhagen tradition of computer science

Since the middle of the 1960s, computer science has been practised in Denmark under Peter Naur's termdatalogy, the science of data processes. Starting at Regenecentralen and the University of Copenhagen, the Copenhagen Tradition of Computer Science has developed its own special characteristics by means of a close connection with applications and other fields of knowledge. The tradition is not least visible in the area of education. Comprehensive project activity is an integral part of the curriculum, thus presenting theory as an aspect of realistic solutions known primarily through actual experience. Peter Naur early recognized the particular educational challenges presented by computer science. His innovations have shown their quality and vitality also at other universities. There is a close connection between computer science training as it has been formed at Copenhagen University, and the view of computer science which has characterized Peter Naur's research. We illustrate how the study of programming and system development conceived as a human activity has been an all-pervasive theme in Naur's work. This approach has set the scene for central research issues in software development which today seem more topical than ever.Dedicated to Peter Naur on the occasion of his 60th birthday     

S6中具有常数K?hler角和常数曲率的极小曲面

本文给出了近Ｋ?ｈｌｅｒ球面Ｓ⁶中具有常数Ｋ?ｈｌｅｒ角和常数曲率的极小曲面的例子,同时证明了两个唯一性定理．     

A Geometrical Proof of Shiota's Theorem on a Conjecture of S. P. Novikov

We give a new proof of Shiota's theorem on Novikov's consecture, which states that the K.P. equation characters Jacobians among all indecomposable principally polarized abelian varieties.     

ON A CONJECTURE OF K. OGIUE FOR KAEHLER HYPERSURFACES

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Classical Solutions of Nonautonomous Riccati Equations Arising in Parabolic Boundary Control Problems, II

An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. We give the proof of some result stated in [AT5], and in addition we prove uniqueness of the Riccati operator, provided its final datum is suitably regular. Accepted 14 October 1998     

Minimax Control of Parabolic Systems with Dirichlet Boundary Conditions and State Constraints

In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations, we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions. Accepted 7 June 1996     

Stable boundary layers in a diffusion problem with nonlinear reaction at the boundary

We prove the existence of nonconstant stable stationary solutions of an evolution problem with a nonlinear reaction acting on the boundary. These solutions present layers at certain points of the boundary. We also study the behavior of these solutions as the small parameter appearing in the equation approaches zero and show some stability properties of the profiles given by these equilibrium solutions.     

A degenerate diffusion problem with dynamical boundary conditions Diffusion problem with dynamical boundary conditions

The purpose of this paper is to study the existence, the uniqueness and the limit in , as of solutions of general initial-boundary-value problems of the form and in a bounded domain with dynamical boundary conditions of the form Received: 5 December 2000 / Revised version: 20 November 2001 / Published online: 4 April 2002     

Approximation of general arch problems by straight beam elements

Summary In this paper, we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration. Compatibility conditions which have to be satisfied at the mesh points are given. These conditions ensure for this method the same order of convergence as usual conforming finite element methods.     

Classical solutions of quasilinear parabolic systems on two dimensional domains

Using results on abstract evolutions equations and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems admit a local, classical solution in the space of p–integrable functions, for some p greater than 1, over a bounded two dimensional space domain. The treatment of such equations in a space of integrable functions enables us to define the normal component of the current across the boundary of any Lipschitz subset. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck’s system.     

Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints

This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control.