Mean-Value Theorem with Small Subdifferentials

We prove a mean-value theorem for lower semicontinuous functions on a large class of Banach spaces which contains the class of Asplund spaces, in particular reflexive Banach spaces and Banach spaces with a separable dual. It involves the lower subdifferential (or contingent subdifferential) and the Fréchet subdifferentials, which are among the smallest subdifferentials known to date. It follows that the estimates which it provides require weak assumptions and are accurate. When the function is locally Lipschitzian, we get a simple statement which refines the Lebourg mean-value theorem.     

Subdifferentials with Autoconjugate Fitzpatrick Function

In the framework of real Banach spaces, the present paper provides a necessary and sufficient condition for the Fitzpatrick function of the subdifferential of a proper lower semicontinuous convex function to be autoconjugate. This enables us to: obtain a new proof of the fact that subdifferentials of indicator and sublinear functions have autoconjugate Fitzpatrick functions; characterize those classes of functions whose subdifferentials fulfill the condition under study in the same special way as indicator and sublinear functions do; prove that, in the one-dimensional case, the functions of these classes are the only ones whose subdifferentials have autoconjugate Fitzpatrick functions, while this is not true in higher dimensions.     

Variational analysis of fully coupled electro‐elastic frictional contact problems

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a foundation. The material behavior is described with a nonlinear electro‐elastic constitutive law. The novelty of the model consists in the fact that the foundation is assumed to be electrically conductive and both the frictional contact and the conductivity on the contact surface are described with subdifferential boundary conditions which involve a fully coupling between the mechanical and electrical variables. We derive a variational formulation of the problem which is in the form of a system coupling two hemivariational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on recent results for inclusions of subdifferential type in Sobolev spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the subdifferential of a submodular function

The author recently introduced a concept of a subdifferential of a submodular function defined on a distributive lattice. Each subdifferential is an unbounded polyhedron. In the present paper we determine the set of all the extreme points and rays of each subdifferential and show the relationship between subdifferentials of a submodular function and subdifferentials, in an ordinary sense of convex analysis, of Lovász's extension of the submodular function. Furthermore, for a modular function on a distributive lattice we give an algorithm for determining which subdifferential contains a given vector and finding a nonnegative linear combination of extreme vectors of the subdifferential which expresses the given vector minus the unique extreme point of the subdifferential.     

Nonlinear second order differential inclusions with a series of convex functions

In this paper, we study a class of second order differential inclusions attached to which the convex potential varies depending on the time, and the boundary conditions are described by the subdifferentials of convex functions. Applying the theory of lower semicontinuous, convex and proper functionals, and fixed points, we give some results about the representation of the subdifferential operators of some convex functionals and the existence of solutions for the boundary value problems.     

Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmooth analysis. Then we study several versions of smooth variational principles in Asplund spaces, provide necessary and sufficient conditions for the validity of such principles, and establish their relationships with certain subdifferential properties of lower semicontinuous functions.     

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

We continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina, The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples (this journal)]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel-Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe’s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained.     

The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values

We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent.     

Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces

Critical point theory is used to show the existence of weak solutions to a quasilinear elliptic differential equation under the functional framework of the Musielak–Sobolev spaces in a bounded smooth domain with Dirichlet boundary condition.     

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACT

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.     

Continuous Outer Subdifferentials in Nonsmooth Optimization

The theory of subdifferentials provides adequate methods and tools to put descent methods for nonsmooth optimization problems into practice. However, in applications it is often difficult to decide on a suitable subdifferential concept to construct a descent method. Therefore, we introduce subdifferentials in terms of their properties to indicate a selection of subdifferentials worth considering. This initials the first part of the construction of a continuous outer subdifferential (COS). Typically, methods based on e.g. the Clarke subdifferential are non-convergent without assumptions like semismoothness on the objective function. In cases in which only supersets of the Clarke subdifferential are known, semismoothness cannot be proved or is even violated. Therefore, in the second part of the construction, a previously selected subdifferential will be expanded to a continuous mapping, if necessary. This is also practicable for upper bounds of the subdifferential of current interest. Finally, based on COS we present a methodology for solving nonsmooth optimization problems. From a theoretical point of view, convergence is established through the construction of COS.

     

Elastic contact problem with Coulomb friction and normal compliance in Orlicz spaces

A static contact problem for inhomogeneous elastic materials is studied with a non-polynomial growth of the elasticity under the Coulomb’s law of dry friction and the normal compliance condition. We demonstrate the results on existence and uniqueness of a solution to an abstract subdifferential inclusion and a variational–hemivariational inequality in the reflexive Orlicz–Sobolev space which are applied to the static elastic frictional problem.     

A justification for the thin film approximation of Stokes flow with surface tension

In the free boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. It is rigorously shown that in this limit the evolution is given by the well-known thin film equation. The main techniques are appropriate scaling and uniform energy estimates in Sobolev spaces of sufficiently high order, based on parabolicity.     

Subdifferential formulas for a class of non-convex infimal convolutions

In this paper, we provide a number of subdifferential formulas for a class of non-convex infimal convolutions in normed spaces. The formulas obtained unify several results on subdifferentials of the distance function and the minimal time function. In particular, we generalize the results obtained recently by Zhang et al.     

Sobolev problem in the complete scale of Banach Spaces

In a bounded domainG ? ? ⁿ , whose boundary is the union of manifolds of different dimensions, we study the Sobolev problem for a properly elliptic expression of order 2m. The boundary conditions are given by linear differential expressions on manifolds of different dimensions. We study the Sobolev problem in the complete scale of Banach spaces. For this problem, we prove the theorem on a complete set of isomorphisms and indicate its applications.     

Diffraction of light revisited

The diffraction of monochromatic light is considered for a plane screen with an open infinite slit by solving the vectorial Maxwell–Helmholtz system in the upper half‐space with the Fourier method. With this approach we can represent each solution satisfying an appropriate energy condition by its boundary fields in the Sobolev spaces H^±1/2. We show that Sommerfeld's theory using a boundary integral equation with Hankel kernels for the so‐called B‐polarization is covered by our approach, but in general it violates a necessary energy condition. Our representation includes also solutions which are not covered by Sommerfeld's theory. Copyright © 2007 John Wiley & Sons, Ltd.     

Subdifferential estimate of the directional derivative,optimality criterion and separation principles

We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate ?-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate ?-subdifferential.     

Decay of the 2D Periodic Stationary Viscous Surface Waves

We consider the evolution of viscous fluids in a 2D horizontally periodic slab bounded above by a free top surface and below by a fixed flat bottom. This is a free boundary problem. The dynamics of the fluid are governed by the incompressible stationary Navier–Stokes equations under the influence of gravity and the effect of surface tension. We develop the global theory of solutions in low regularity Sobolev spaces for small data by nonlinear energy estimates.     

Characterizations of the solution set for non-essentially quasiconvex programming

Characterizations of the solution set in terms of subdifferentials play an important role in research of mathematical programming. Previous characterizations are based on necessary and sufficient optimality conditions and invariance properties of subdifferentials. Recently, characterizations of the solution set for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential are studied by the authors. Unfortunately, there are some examples such that these characterizations do not hold for non-essentially quasiconvex programming. As far as we know, characterizations of the solution set for non-essentially quasiconvex programming have not been studied yet. In this paper, we study characterizations of the solution set in terms of subdifferentials for non-essentially quasiconvex programming. For this purpose, we use Martínez–Legaz subdifferential which is introduced by Martínez–Legaz as a special case of c-subdifferential by Moreau. We derive necessary and sufficient optimality conditions for quasiconvex programming by means of Martínez–Legaz subdifferential, and, as a consequence, investigate characterizations of the solution set in terms of Martínez–Legaz subdifferential. In addition, we compare our results with previous ones. We show an invariance property of Greenberg–Pierskalla subdifferential as a consequence of an invariance property of Martínez–Legaz subdifferential. We give characterizations of the solution set for essentially quasiconvex programming in terms of Martínez–Legaz subdifferential.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.