Expressions for subdifferential and optimization problems defined by nonsmooth homogeneous functions

In this paper, we prove a theoretical expression for subdifferentials of lower semicontinuous and homogeneous functions. The theoretical expression is a generalization of the Euler formula for differentiable homogeneous functions. As applications of the generalized Euler formula, we consider constrained optimization problems defined by nonsmooth positively homogeneous functions in smooth Banach spaces. Some results concerning Karush–Kuhn–Tucker points and necessary optimality conditions for the optimization problems are obtained.     

A Variational Formulation of Analytical Mechanics in an Affine Space

Variational formulations of statics and dynamics of mechanical systems controlled by external forces are presented as examples of variational principles.     

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

We show how ‘test’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger’s constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger’s constant in terms of the inradius of a plane domain. Received: 13 June 2005     

A Zubov’s method for stochastic differential equations

We consider a stochastic differential equation with an asymptotically stable equilibrium point. We show that the domain of attraction of the equilibrium, i.e. the set of points which are attracted with positive probability to it, can be characterized by the solution of a suitable partial differential equation.     

Homogenization and penalization of functional first-order PDE

In this paper we study the asymptotic behavior of viscosity solutions for a functional partial differential equation with a small parameter as the parameter tends to zero. We study simultaneous effects of homogenization and penalization in functional first-order PDE. We establish a convergence theorem in which the limit equation is identified with some first order PDE.     

Stability and perturbations of the domain for the first eigenvalue of the 1-Laplacian

We study the dependence of the first eigenvalue of the 1-Laplacian with respect to perturbations of the domain. We provide results ranging from general type of perturbations to regular perturbations by diffeomorphisms. Received: 31 March 2005     

On the existence of solutions to generalized vector variational-like inequalities

In this paper, we consider a vector version of generalized Minty's lemma and obtain existence theorems of solutions for two kinds of vector variational-like inequalities.     

On general properties of phenomenological and state coefficients for isotropic viscoanelastic media

The aim of this work is the study of general properties of phenomenological and state coefficients in a thermomechanical model for viscoanelastic media proposed by one of us (VC), in which cross-effect between viscous and inelastic flow is neglected. In particular, after the specification of the low and high frequencies region, under physical considerations characterized by limiting the values of some functions connected to mechanical phenomena, generally valid behaviour for phenomenological and state coefficients as function of the frequency are deduced. It has been possible, from the entropy production, to determine some characteristics of the coefficients as function of frequency in order that this principle is satisfied. The application of the so obtained results to a polymeric material as PolyIsobutilene in these regions in which, in a first approximation, the medium shows a linear behaviour, confirms our deductions.     

Optimal Shape Design of Blazed Diffraction Gratings

The problem of designing a periodic interface between two materials in such a way that time-harmonic waves diffracted from the interface have a specified far-field pattern is studied. A minimization problem for the interface is formulated, and it is shown that solutions of constrained bounded variation exist. The differentiability of the cost functional is then studied, with no restrictions on the smoothness of the interface. Some computational issues are discussed, and finally the results of some numerical experiments are presented. Accepted 3 February 1998     

Two Extensions to Finsler's Recurring Theorem

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R ⁿ : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R ⁿ → R ^k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y R ⁿ for some λ∈ R ^k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R ⁿ by a real vector space X . Accepted 22 February 1998