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A Note on Quasiconvexity and Rank-one Convexity for 2$\times$2 Matrices
Quasiconvexity 2$\times$2 Matrices Rank-one Convexity
2009/1/22
We report on an attempt to find a counterexample to the statement that rank-one convexity does not imply quasiconvexity in the case of 2$\times$2 matrices. The failure of such attempt is a consequence...
On Schur-Ostrowski Type Theorems for Group Majorizations
preordering cone preordering majorization group majorization group induced cone ordering convex cone dual cone, isotone function Schur convex function fundamental region reflection group Schur-Ostrowski's condition
2009/1/22
In this paper the problem of the isotonicity of a function with respect to a group majorization is discussed. For a group induced cone ordering the isotonicity is characterized via some conditions of...
On a Variational Inequality Containing a Memory Term with an Application in Electro-Chemical Machining
Evolutionary variational inequality existence uniqueness regularity and time evolution of the solution convex sets penalization method electro-chemical machining process
2009/1/22
An obstacle problem with a memory term is studied in the framework of the variational inequality theory. Applying a fixed point argument and a convergence result for convex sets the existence and uniq...
On a New Result on the Existence of Zeros due to Ricceri
Zeros dual space Ricceri arbitrary topological spaces
2009/1/22
The purpose of this short paper is to extend a recent result of Ricceri, on the existence of zeros of functions with values in a dual space, to the existence of solutions to inclusions with values in ...
Spectrum Stability of an Elliptic Operator to Domain Perturbations
Domain perturbation eigenvalue elliptic operator shape continuity
2009/1/22
This paper raises the question of the stability of the full spectrum of an elliptic operator in divergence form (with homogeneous Dirichlet boundary conditions) with respect to domain perturbations w...
Non-Coercive Variational Problems with Constraints on the Derivatives
Non-Coercive Variational Problems Derivatives Constraints
2009/1/22
We establish a necessary and sufficient condition for the existence of the minimum of the functional $\dis{\int_0^1} f(t,v^\prime(t))dt$ in the class ${\cal W}_d^p=\{v\in W^{1,p}([0,1]): v(0)=0, v(1)=...
The Barrier Cone of a Convex Set and the Closure of the Cover
convex set barrier cone recession cone cover polar cone
2009/1/22
For an arbitrary non-empty closed convex set $A$ in $\mathbb{R}^n$, we prove that the polar of the difference between the barrier cone $\mathbb{B}(A)$ and its interior $\text{int } \mathbb{B} (A)$ co...
A Note on the Closedness of the Convex Hull and Its Applications
Convex Hull Applications spannability
2009/1/22
This paper answers the following question motivated by the problem of spannability of functions. When is the convex hull of an unbounded (closed) set closed? We provide necessary and sufficient condit...
Painleve-Kuratowski convergence results are obtained, under sufficient conditions of minimal character, for approximate Nash equilibria of two sequences of real valued functions. Moreover, an applicat...
Invariants of Pairs of Compact Convex Sets
Pairs of convex sets sublinear function quasidifferential calculus
2009/1/22
In a recent paper P. Diamond, P. Kloeden, A. Rubinov and A. Vladimirov [3] investigated comperative properties of three different metrics in the space of pairs of compact convex sets. These metrics de...
BV Functions with Respect to a Measure and Relaxation of Metric Integral Functionals
Bounded variation functions Radon measures Relaxation Duality Integral representation
2009/1/22
We introduce and study the space of bounded variation functions with respect to a Radon measure $\mu$ on $\mathbb{R}^N$ and to a metric integrand $\varphi$ on the tangent bundle to $\mu$. We show that...
Relationship Between Dynamic Programming and the Maximum Principle under State Constraints
Dynamic Programming Maximum Principle State Constraints
2009/1/22
Bellman's dynamic programming and Pontryagin's maximum principle are two basic tools for studying optimal control theory. We consider the optimal control problem under state constraints and examine th...
Dykstra’s Algorithm as the Nonlinear Extension of Bregman's Optimization Method
Bregman projection convex programming Dykstra's algorithm
2009/1/22
We show that Dykstra's algorithm with Bregman projections, which finds the Bregman projection of a point onto the nonempty intersection of finitely many closed convex sets, is actually the nonlinear e...
Dynamical Shape Control in Non-cylindrical Navier-Stokes Equations
Dynamical Shape Control Non-cylindrical Navier-Stokes Equations
2009/1/22
This paper deals with a dynamical shape control problem. The state equations are the non-cylindrical Navier-Stokes equations with a non-homogenous Dirichlet condition. The goal is to compute a necessa...
Boundary Homogenization for a Quasi-Linear Elliptic Problem with Dirichlet Boundary Conditions Posed on Small Inclusions Distributed on the Boundary
Boundary Homogenization Dirichlet Boundary Conditions Quasi-Linear Elliptic Problem
2009/1/22
We describe the asymptotic behaviour of the solution of a quasi-linear elliptic problem posed in a domain of $\R^n$, $n\geq 3$ and with homogeneous Dirichlet boundary conditions imposed on small zones...