Two Positive Solutions of a Quasilinear Elliptic Dirichlet Problem

For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C ¹–functional on H₀¹(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type.     

Controllability properties of constrained linear systems

In this paper, we present results on constrained controllability for linear control systems. The controls are constrained to take values in a compact set containing the origin. We use the results on reachability properties discussed in Ref. 1.We prove that controllability of an arbitrary pointp inR ⁿ is equivalent to an inclusion property of the reachable sets at certain positive times. We also develop geometric properties ofG, the set of all nonnegative times at whichp is controllable, and ofC, the set of all controllable points. We characterize the setC for the given system and provide additional spectrum-dependent structure.We show that, for the given linear system, several notions of constrained controllability of the pointp are the same, and thus the setC is open. We also provide a necessary condition for small-time (differential or local) constrained controllability ofp.This work was supported in part by NSF Grant ECS-86-09586.     

Extrema Constrained by a Family of Curves and Local Extrema

This paper considers the connections between the local extrema of a function f:DR and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C ¹ or C ²-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C ¹ or C ²-curves. Surprisingly, what is true for C ¹-curves fails to be true in part for C ²-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.     

Second-order conditions in c1, 1 optimization with applications

Second-order necessary conditions for inequality and equality constrained C^{1, 1} optimization problems are derived. A constraint qualification condition which uses the recent generalized second-order directional derivative is employed to obtain these conditions. Various second-order sufficient conditions are given under appropriate conditions on the generalized second-order directional derivative in a neighborhood of a given point. An application of the secondorder conditions to a new class of nonsmooth C^{1, 1} optimization problems with infinitely many constraints is presented.     

Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities

We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex). Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993)) on H¹₀ versus C¹₀ minimizers of a C¹-functional.     

Sommets et normales concourantes des courbes convexes de largeur constante et singularités des hérissons

LetC be a convex curve of constant width and of classC ₄ ⁺ . It is known thatC has at least 6 vertices and its interior contains either a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals. If all its vertices are nondegenerate, then: (i)C has exactly 6 vertices if, and only if, its evolute is the boundary of a topological disc through each interior point of which pass at least 6 normals; (ii) ifC has more than 6 vertices, then there exists an open set of points through each of which pass at least 10 normals. The proof: (i) expresses the number of normals passing through a point as a function of the index with respect to the evolute; (ii) relates this index to the number of singularities of the evolute (i.e. of vertices). Furthermore, we give formulas for counting singularities of generic hedgehogs in ℝ² and ℝ³.

     

On the h-vector of a cohen-macaulay domain in positive characteristic

Here we study the Hilbert function of a Cohen-Macaulay homogeneous domain over an algebraically closed field of positive characteristic. The main tool (and an essential part of the main geometrical results) is the study of the Hilbert function of a general hyperplane section X?P ^r of an integral curve C?P ^r+1 , which is pathological in some sense. In §1, we study the case when Cis a strange curve, i.e., all tangent lines to Cat its simple points pass through a fixed point υ∈P ^r+1 . In §2, we give more refined results under the assumption that the Trisecant Lemma fails for C, i.e., any line spanned by two points of Ccontains one more point of C.     

Pairs of positive solutions for nonlinear elliptic equations with the <Emphasis Type="Italic">p</Emphasis>-Laplacian and a nonsmooth potential

In this paper we examine a nonlinear elliptic problem driven by the p-Laplacian differential operator and with a potential function which is only locally Lipschitz, not necessarily C¹ (hemivariational inequality). Using the nonsmooth critical point theory of Chang, we obtain two strictly positive solutions. One solution is obtained by minimization of a suitable modification of the energy functional. The second solution is obtained by generalizing a result of Brezis-Nirenberg about the local C¹₀-minimizers versus the local H¹₀-minimizers of a C¹-functional. Mathematics Subject Classification (2000) 35J50, 35J85, 35R70     

Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay

Standard software based on the collocation method for differential equations delivers a continuous approximation (called the collocation solution) which augments the high order discrete approximate solution that is provided at mesh points. This continuous approximation is less accurate than the discrete approximation. For ‘non-standard’ Volterra integro-differential equations with constant delay, that often arise in modeling predator-prey systems in Ecology, the collocation solution is C ⁰ continuous. The accuracy is O(h ^s+1) at off-mesh points and O(h ^2s ) at mesh points where s is the number of Gauss points used per subinterval and h refers to the stepsize. We will show how to construct C ¹ interpolants with an accuracy at off-mesh points and mesh points of the same order (2s). This implies that even for coarse mesh selections we achieve an accurate and smooth approximate solution. Specific schemes are presented for s=2, 3, and numerical results demonstrate the effectiveness of the new interpolants.     

The second order optimality conditions for nonlinear mathematical programming with C 1,1 data

To find nonlinear minimization problems are considered and standard C ²-regularity assumptions on the criterion function and constrained functions are reduced to C ^1,1-regularity. With the aid of the generalized second order directional derivative for C ^1,1 real-valued functions, a new second order necessary optimality condition and a new second order sufficient optimality condition for these problems are derived.     

Finite sets on curves and surfaces

A complete proof is given for Schnirelmann’s theorem on the existence of a square inC ² Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface inR ⁿ,C ³-diffeomorphic toS ⁿ⁻¹, there exist 2n points which are the vertices of a regular 2 ⁿ -cellC _n. 2. Every planeC′ Jordan curve can beC′ approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (anglesπ not excluded). 3. On every planeC ² curve there exist 5 distinct points which are the vertices of an axially symmetric pentagon with given base anglesa, π/2≦a<π. (The angle at the vertex on the axis of symmetry might beπ). Research supported by Grant AF-AFOSR-664-64, Air Force Office of Scientific Research.     

The C 1 wavelet method for numerical solutions of the Neumann problem

In this article we use the C ¹ wavelet bases on Powell-Sabin triangulations to approximate the solution of the Neumann problem for partial differential equations. The C ¹ wavelet bases are stable and have explicit expressions on a three-direction mesh. Consequently, we can approximate the solution of the Neumann problem accurately and stably. The convergence and error estimates of the numerical solutions are given. The computational results of a numerical example show that our wavelet method is well suitable to the Neumann boundary problem.     

Generalized Hessian matrix and second-order optimality conditions for problems withC 1,1 data

In this paper, we present a generalization of the Hessian matrix toC ^1,1 functions, i.e., to functions whose gradient mapping is locally Lipschitz. This type of function arises quite naturally in nonlinear analysis and optimization. First the properties of the generalized Hessian matrix are investigated and then some calculus rules are given. In particular, a second-order Taylor expansion of aC ^1,1 function is derived. This allows us to get second-order optimality conditions for nonlinearly constrained mathematical programming problems withC ^1,1 data.     

TheN-membranes problem

We study the equilibrium position ofN elastic membranes attached to rigid supports and submitted to the action of forces. They are constrained because they cannot pass through each other. As in the case of the obstacle problem, the solution fails to beC ² and thus fails to be classical, so we provide some new regularity results in different larger spaces using an iterative penalization technique.On leave from Departement de Mathematiques, Universite de Metz, Ile du Saulcy, 57045 Metz-Cedex, FranceOn leave from Dipartamento di Matematica, Universita di Pisa, 56100 Pisa, Italy     

On Ω-blow-ups in the simplest <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth skew products of interval mappings

The authors prove a criterion (necessary and sufficient condition) for the emergence of the C ⁰-Ω-blow-up for C ¹-smooth skew products of interval mappings with closed set of periodic points. An example of the mapping with given properties that admits the C ⁰-Ω-blow-up is presented. It is proved that the C ¹-Ω-blow-up is impossible for mappings of such a type (in the space of C ¹-smooth skew products of interval mappings). It is proved that there is no one-parameter family of C ¹-smooth skew products of interval mappings with closed set of periodic points C ¹-smoothly depending on the parameter in which from one fixed point, periodic orbits with periods 2 and 4 simultaneously arise. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ? F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A _2n n∈N, E₆ and E₈. We give conditions for C to pass through rational singularities of types. A _2k+1 k∈Z⁺ D_n n≥4 and E₇, (0.8).     

Detectability of critical points of smooth functionals from their finite-dimensional approximations

Given a critical point of a C²-functional on a separable Hilbert space, we obtain sufficient conditions for it to be detectable (i.e. ‘visible’) from finite-dimensional Rayleigh-Ritz-Galerkin (RRG) approximations. While examples show that even nondegenerate critical points are, without any further restriction, not visible, we single out relevant classes of smooth functionals, e.g. the Hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations, such that their nondegenerate critical points are visible from their RRG approximations.     

Fixed points of C2 maps

S.N. Chow and J.A. Yorke have proposed in abstract terms an algorithm for computing fixed points of C² maps that is globally convergent with probability one. A numerical implementation of that algorithm is presented here, where careful attention has been paid to computational efficiency, accuracy, and robustness. Convergence proofs for the numerical algorithm require differential geometry, and are given elsewhere. FORTRAN subroutines are given and explained in detail, and some typical numerical results are presented. It is shown how to modify the subroutines to compute zeros and handle some large sparse problems.     

Approximation by cubicC 1-splines on arbitrary triangulations

Summary In this paper, we present an efficient representation for bivariate piecewise cubicC ¹-splines on arbitrary triangulations. A numerical method is discussed for computing the dimension of the spaceS ₃ ¹ () of these splines. We consider subspaces ofS ₃ ¹ () satisfying certain boundary conditions. Some applications are given where piecewise cubicC ¹-functions are used to solve interpolation problems and least squares approximation problems.     

Regularity of Irregular Subdivision

We study the smoothness of the limit function for one-dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural generalization of the four-point scheme introduced by Dubuc and Dyn, Levin, and Gregory, we show that, under some geometric restrictions, the limit function is always C ¹ ; under slightly stronger restrictions we show that the limit function is almost C ² , the same regularity as in the regularly spaced case. May 27, 1997. Date revised: March 10, 1998. Date accepted: March 28, 1998.