On the structure of quasiconvex hulls

We define the set K_{q,e ⊂ K} of quasiconvex extreme points for compact sets K ⊂ M^N×n and study its properties. We show that K_q,e is the smallest generator of Q(K)-the quasiconvex hull of K, in the sense that Q(K_q,e) = Q(K), and that for every compact subset W ⊂ Q(K) with Q(W) = Q(K), K_q,e ⊂ W. The set of quasiconvex extreme points relies on K only in the sense that . We also establish that K_e ⊂ K_q,e, where K_e is the set of extreme points of C(K)-the convex hull of K. We give various examples to show that K_q,e is not necessarily closed even when Q(K) is not convex; and that for some nonconvex Q(K), K_q,e = K_e. We apply the results to the two well and three well problems studied in martensitic phase transitions.     

The quasiconvex hull for the five-gradient problem

In [8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying ${Du \in K}$ . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each ${F \in {\rm int}\,K^{rc}}$ there exists a Lipschitz mapping u satisfying $$Du \in K\quad{\rm and}\quad u(x) = Fx\,{\rm for}\,x\,\in\,{\partial} \Omega \,.$$ Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal.     

On quasiconvex hulls in symmetric matrices

In this paper we study the quasiconvex hull of compact sets of symmetric 2×2 matrices. We are interested in situations where the quasiconvex hull can be separated into smaller independent pieces. Our main result is a geometric criterion which is sufficient for the quasiconvex hull of the union of two compact sets K₁K₂ to separate in the sense that . The key point in the proof is a kind of directional maximum principle for second order elliptic equations in the plane in non-divergence form with measurable coefficients.     

On extremal points of quasiconvex functions

In this paper we examine the linear sectionwise relative minimums of a quasiconvex function and give a sufficient condition for quasiconvex functions to have a strict global minimum on an open convex set.     

On the regularity of the minima of quasiconvex integrals

We prove in this paper theC ^∞ regularity for a “very strict” local minimum of classC _loc ^ρ , ρ>3, of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC ₁ (x)>0,C ₂ (x)>0,1≥ε(x)>0, such that for all real ϕ∈c ₀ ^∞ (K). This work is supported by the National Natural Science Foundation of China and the Fok Ying Tung Education Foundation.     

Robust multigrid method for the planar linear elasticity problems

We consider the solution of the system of equations that arise from the higher order conforming finite element (Scott–Vogelius element) discretizations of the boundary value problems associated with the differential operator −ρ ² Δ − κ ²∇div, where ρ and κ are nonzero parameters. Robust multigrid method is constructed, i.e., the convergence rate of multigrid method is optimal with respect to the mesh size, the number of levels, and weights on the two terms in the aforementioned differential operator.

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Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices

In a recent paper, G. Czédli and E. T. Schmidt present a structure theorem for planar semimodular lattices. In this note, we present an alternative proof.     

On the infimum of a quasiconvex vector function over an intersection

We give sufficient conditions for the infimum of a quasiconvex vector function f over an intersection \(\bigcap_{i\in I}R_{i}\) to agree with the supremum of the infima of f over the R _i ’s.     

Second-order locking-free nonconforming elements for planar linear elasticity

In this paper, we present two nonconforming finite elements for the pure displacement planar elasticity problem. Both of them are locking-free and have two order of convergence. Some numerical results attest the validity of our theoretical analysis.     

On the cotorsion hull in torsion theory

The concept of cotorsion was first introduced in the category of Abelian groups (Fuchs [l] ). Matlis [5], studied the cotorsion modules over integral domains. Henderson and Orzech [4], Fuchs [2], and Mines [6], replaced the classical notion of torsion by a torsion theory (T,F) on R-mod, where R is not necessarily commutative ring. In this paper we find conditions on the torsion theory in order to get a T-cotorsion hull for every module. This generalizes the result of Fuchs [2].     

On the Hamiltonian structure of the planar steady water-wave problem with vorticity

We consider the stream-function formulation of the hydrodynamic problem for steady rotational water waves both with and without surface tension. A natural Lagrangian formulation is presented from which (different) Hamiltonian formulations for the two cases are derived by duality theory in the spirit of the Legendre–Fenchel transform. The treatment is systematic and clarifies a recent ad hoc approach by Kozlov and Kuznetsov [7].     

On monotone hull operations

We extend results of Elekes and Máthé on monotone Borel hulls to an abstract setting of measurable space with negligibles. This scheme yields the respective theorems in the case of category and in the cases associated with the Mendez σ‐ideals on the plane. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On Karamardian's theorem about lower semicontinuous strictly quasiconvex functions

Summary A well-known theorem ofKaramardian [1967] says: A lower semicontinuous and strictly quasiconvex real-valued function defined on a convex subset ofR ⁿ is quasiconvex. A proof is given for the following generalization: A connected quasiorder valued function, defined on a convex subset of a normed real linear space, which is lower semicontinuous and strictly midpointquasiconvex is actually both quasiconvex and strictly quasiconvex. A number of similar results are proved interrelating the concepts of midpoint-quasiconvexity, rational quasiconvexity, quasiconvexity and their strict counterparts.

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Zusammenfassung Ein bekannter Satz vonKaramardian [1967] lautet: Eine strikt quasikonvexe und nach unten halbstetige reellwertige Funktion auf einer konvexen Teilmenge desR ⁿ ist quasikonvex. Es wird folgende Verallgemeinerung bewiesen: Eine strikt mittelpunktsquasikonvexe und nach unten halbstetige konnexquasiordnungswertige Funktion auf einer konvexen Teilmenge eines normierten reellen linearen Raumes ist quasikonvex und strikt quasikonvex. Ferner wird eine Anzahl ähnlicher Aussagen bewiesen, die die Begriffe Mittelpunktsqausikonvexität, rationale Quasikonvexität und Quasikonvexität sowie deren strikte Entsprechungen miteinander in Beziehung setzen.

     

On the connectedness of the efficient set for strictly quasiconvex vector minimization problems

In this paper, we investigate the connectedness of the efficient solution set for vector minimization problems defined by a continuous vector-valued strictly quasiconvex functionf=(f ₁,...,f _m ) _T and a convex compact setX. It is shown that the efficient solution set is connected if one component off is strongly quasiconvex onX.The author would like to thank Professor H. P. Benson and the referees for many valuable comments and for pointing out some errors in the previous draft.Formerly, Assistant, Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai, China.     

On the circular hull property in normed planes

We extend the notion of circular hull to arbitrary normed planes and prove that a compact, convex set of constant Minkowskian width has the circular hull property in such a plane. Also we show how this property is related to the so called weak circular intersection property.     

On sufficient optimality conditions for a quasiconvex programming problem

Some remarks are made on a paper by Bector, Chandra, and Bector (see Ref. 1) concerning the Fritz John and Kuhn-Tucker sufficient optmality conditions as well as duality theorems for a nonlinear programming problem with a quasiconvex objective function.This research was supported by the Italian Ministry of University Scientific and Technological Research.     

On the structure of the spectrum for the elasticity problem in a body with a supersharp spike

We establish that the continuous spectrum of the Neumann problem for the system of elasticity equations occupies the entire closed positive real semiaxis in the case that a three-dimensional body with a sharp-spiked cusp whose cross-section contracts to a point with the velocity O(r ^1+γ), where r is the distance to the vertex of the spike and γ > 1 is the sharpness exponent.