Volume 13, Issue 2

This article is concerned with the existence result of the unilateral problem associated to the equations of the type Au – div φ ( u ) = ƒ ∈ L 1 (Ω), where A is a Leray-Lions operator having a growth not necessarily of polynomial type and φ ∈ C 0 (ℝ, ℝ N ).

The present paper provides first and second-order characterizations of a radially lower semicontinuous strictly pseudoconvex function ƒ : X → ℝ defined on a convex set X in the real Euclidean space ℝ n in terms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [J. Econom. Theory 25: 397–420, 1981]. A new implication appears in our conditions. Connections with other classes of functions are also derived.

This paper presents a differential criterion of n dimensional geometrically convex functions, and gives some applications.

We discuss the weak form of the Ramberg/Osgood equations (also known as the Norton/Hoff model) for nonlinear elastic materials on a 3-dimensional domain and show that the stress tensor is Hölder continuous on an open subset whose complement is of Lebesgue-measure zero. We also give an estimate for the Hausdorff-dimension of the singular set.

We investigate a class of over-determined parabolic problems involving a non-constant boundary condition. The Weinstein's technique known for the elliptic problems is extended to the parabolic one by means of auxiliary functions and Green classical formula.

We prove that if a pair 〈 I , J 〉 of ccc, translation invariant σ -ideals on 2 ω has the Fubini Property, then I = J . This leads to a slightly improved exposition of a part of the Farah-Zapletal proof of an invariant version of their theorem which characterizes the measure and category σ -ideals on 2 ω as essentially the only ccc definable σ -ideals with Fubini Property.

In this paper, we construct γ -regular Cl n -minimal function systems in , the generalized Bergman space of Cl n -valued functions in the Sobolev space which are used in the best way to approximate null solutions of the in-homogeneous Dirac operator.

We introduce “probabilistic” and “stochastic Hilbertian structures”. These seem to be a suitable context for developing a theory of “quantum Gaussian processes”. The Schauder system is utilised to give a Lévy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given.

The solvability of the generalized weak vector implicit variational inequality problem, generalized strong vector implicit variational inequality problem and generalized vector variational inequality problem are proved by using a generalized Fan's KKM theorem. Our results extend and unify corresponding results of other authors.