Involutorische Automorphismen ebener konvexer Mengen

 A bijection of the compact convex set with is called a reflection of K, if σ maps convex subsets of K into convex subsets. Conditions are stated unter which the existence of few reflections imply that is an ellipse. Received 4 March, 1998     

Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity

The aim of this paper is to establish existence results for some variational-like inequality problems involving set-valued maps, in reflexive and nonreflexive Banach spaces. When the set K, in which we seek solutions, is compact and convex, we no dot impose any monotonicity assumptions on the set-valued map A, which appears in the formulation of the inequality problems. In the case when K is only bounded, closed, and convex, certain monotonicity assumptions are needed: We ask A to be relaxed η-α monotone for generalized variational-like inequalities and relaxed η-α quasimonotone for variational-like inequalities. We also provide sufficient conditions for the existence of solutions in the case when K is unbounded, closed, and convex.     

Quasiconvex Minimization on a Locally Finite Union of Convex Sets

Extending the approach initiated in Aussel and Hadjisavvas (SIAM J. Optim. 16:358–367, 2005) and Aussel and Ye (Optimization 55:433–457, 2006), we obtain the existence of a local minimizer of a quasiconvex function on the locally finite union of closed convex subsets of a Banach space. We apply the existence result to some difficult nonconvex optimization problems such as the disjunctive programming problem and the bilevel programming problem. Dedicated to Jean-Pierre Crouzeix on the occasion of his 65th birthday. The authors thank two anonymous referees for careful reading of the paper and helpful suggestions. The research of the second author was partially supported by NSERC/Canada.     

On Global Attractors of Multivalued Semiflows Generated by the 3D Bénard System

In this paper we prove the existence of solutions for the 3D Bénard system in the class of functions which are strongly continuous with respect to the second component of the vector (that is, the one corresponding to the parabolic equation). We construct then a multivalued semiflow generated by such solutions and obtain the existence of a global φ −attractor for the weak-strong topology. Moreover, a family of multivalued semiflows is defined on suitable convex bounded subsets of the phase space, proving for them the existence of a global attractor (which is the same for every semiflow of the family) for the weak-strong topology.     

On the Metric Projection Operator and Its Applications to Solving Variational Inequalities in Banach Spaces

In this paper, we investigate the characteristics of the metric projection operator P _K : B → K, where B is a Banach space with dual space B?, and K is a nonempty closed convex subset of B. Then we apply its properties to study the existence of solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.     

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

We consider the wave equation on an interval of length 1 with an interior damping at ξ. It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3–4):215–240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction x = \fracpq,\xi=\frac{p}{q}, where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197–243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337–365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.     

The Existence and Uniqueness of the Solution for Neutral Stochastic Functional Differential Equations with Infinite Delay in Abstract Space

The main aim of this paper is to prove the existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay, which the initial data belong to the phase space ℬ((−∞,0];ℝ ^d ). The vital work of this paper is to extend the initial function space of the paper (Wei and Wang, J. Math. Anal. Appl. 331:516–531, 2007) and give some examples to show that the phase space ℬ((−∞,0];ℝ ^d ) exists. In addition, this paper builds a Banach space ℳ²((−∞,T],ℝ ^d ) with a new norm in order to discuss the existence and uniqueness of the solution for such equations with infinite delay.     

Second Order Nonlinear Evolution Inclusions Ⅰ： Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.     

Convex integration with constraints and applications to phase transitions and partial differential equations

We study solutions of first order partial differential relations Du∈K, where u:Ω⊂ℝ ⁿ →ℝ ^m is a Lipschitz map and K is a bounded set in m×n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of Du and second we replace Gromov’s P-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite. Received April 23, 1999 / final version received September 11, 1999     

Modified Proximal-Point Algorithm for Maximal Monotone Operators in Banach Spaces

We introduce an iterative sequence for finding the solution to 0∈T(v), where T : E⇉E ^* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative procedure is a combination of iterative algorithms proposed by Kohsaka and Takahashi (Abstr. Appl. Anal. 3:239–249, 2004) and Kamamura, Kohsaka and Takahashi (Set-Valued Anal. 12:417–429, 2004). We prove a strong convergence theorem and a weak convergence theorem under different conditions respectively and give an estimate of the convergence rate of the algorithm. An application to minimization problems is given. This work was partially supported by the National Natural Sciences Grant 10671050 and the Heilongjiang Province Natural Sciences Grant A200607. The authors thank the referees for useful comments improving the presentation and Professor K. Kohsaka for pointing out Ref. 7.     

Unavoidable Crossings in a Thinnest Plane Covering with Congruent Convex Disks

Two convex disks K and L in the plane are said to cross each other if the removal of their intersection causes each disk to fall into disjoint components. Almost all major theorems concerning the covering density of a convex disk were proved only for crossing-free coverings. This includes the classical theorem of L. Fejes Tóth (Acta Sci. Math. Szeged 12/A:62–67, 1950) that uses the maximum area hexagon inscribed in the disk to give a significant lower bound for the covering density of the disk. From the early seventies, all attempts of generalizing this theorem were based on the common belief that crossings in a plane covering by congruent convex disks, being counterproductive for producing low density, are always avoidable. Partial success was achieved not long ago, first for “fat” ellipses (A. Heppes in Discrete Comput. Geom. 29:477–481, 2003) and then for “fat” convex disks (G. Fejes Tóth in Discrete Comput. Geom. 34(1):129–141, 2005), where “fat” means of shape sufficiently close to a circle. A recently constructed example will be presented here, showing that, in general, all such attempts must fail. Three perpendiculars drawn from the center of a regular hexagon to its three nonadjacent sides partition the hexagon into three congruent pentagons. Obviously, the plane can be tiled by such pentagons. But a slight modification produces a (non-tiling) pentagon with an unexpected covering property: every thinnest covering of the plane by congruent copies of the modified pentagon must contain crossing pairs. The example has no bearing on the validity of Fejes Tóth’s bound in general, but it shows that any prospective proof must take into consideration the existence of unavoidable crossings.     

On Economic Equilibrium Type Problems with Applications

The existence solution for a class of economic equilibrium type problems in reflexive Banach space is considered. New results concerning the variational inequality approach to Arrow–Debreu model of economic equilibrium introduced in Naniewicz (Math Oper Res 32:436–466, 2007) are found and applied to ensure the existence of Pareto optimal solutions for a class of multiobjective optimization problems with so-called budget-like constraints. To achieve this goal, the theory of pseudo-monotone multivalued mappings combined with some fixed point technique for multivalued mappings with nonconvex values is used.     

Global existence of weak solutions for the Navier�CStokes equations with capillarity on the half-line

In this paper, we construct the global weak solutions to the initial-boundary problem for the Navier–Stokes system with capillarity in the half space \mathbbR₊¹{\mathbb{R}_+^1}. The result extends Eugene Tsyganov’s existence theorem which considered the problem in the finite region published in J. Differential Equaions 245:3936–3955, 2008.     

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565–1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237–2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn–Hilliard equation.     

Tube Formulas and Complex Dimensions of Self-Similar Tilings

We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubularzeta function and hence develop a tube formula for self-similar tilings in ℝ^d. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝ ^d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,…,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).     

Convex decompositions in the plane and continuous pair colorings of the irrationals

We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ² which is not presentable as a countable union of convex sets satisfies the following dichotomy:

Covering Convex Bodies by Cylinders and Lattice Points by Flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1≤k≤d−1. K. Bezdek and A.E. Litvak are partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

An Inexact Proximal-Type Algorithm in Banach Spaces

In this paper, we investigate the strong convergence of an inexact proximal-point algorithm. It is known that the proximal-point algorithm converges weakly to a solution of a maximal monotone operator, but fails to converge strongly. Solodov and Svaiter (Math. Program. 87:189–202, 2000) introduced a new proximal-type algorithm to generate a strongly convergent sequence and established a convergence result in Hilbert space. Subsequently, Kamimura and Takahashi (SIAM J. Optim. 13:938–945, 2003) extended the Solodov and Svaiter result to the setting of uniformly convex and uniformly smooth Banach space. On the other hand, Rockafellar (SIAM J. Control Optim. 14:877–898, 1976) gave an inexact proximal-point algorithm which is more practical than the exact one. Our purpose is to extend the Kamimura and Takahashi result to a new inexact proximal-type algorithm. Moreover, this result is applied to the problem of finding the minimizer of a convex function on a uniformly convex and uniformly smooth Banach space. L.C. Zeng’s research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in Shanghai. J.C. Yao’s research was partially supported by the National Science Council of the Republic of China.     

Complete classification of the positive solutions of −Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We study the equation −Δu + u ^q = 0, q > 1, in a bounded C ² domain Ω ⊂ ℝ ^N . A positive solution of the equation is moderate if it is dominated by a harmonic function and σ-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the subcritical case, 1 < q <, q _c = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More recently, Dynkin proved, by probabilistic methods, that this remains valid in the supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this paper, we prove that for all q ≥ q _c , every positive solution is σ-moderate. We use purely analytic techniques, which apply to the full supercritical range. The main tools come from linear and non-linear potential theory. Combined with previous results, our result establishes a one-to-one correspondence between positive solutions and their boundary traces in the sense of [36].