A remark on Serrin’s Theorem

An approximation theorem for families of weakly coercive convex functions by means of countably many smooth families of convex functions is proved. As a consequence, the classical “three-fold” lower semicontinuity theorem for integral functionals of the Calculus of Variations by James Serrin is unified in one single statement.     

Modified continuity and a generalisation of Michael's selection theorem

We modify the definitions of continuity and lower semicontinuity for single-valued mappings and upper and lower semicontinuity for set-valued mappings. For single-valued mappings we have a generalisation of Osgood's theorem and for set-valued mappings we have an extension of Fort's theorem and a generalisation of Michael's selection theorem producing a densely defined selection with a natural continuity property relative to the domain.     

Weak lower semicontinuity of integral functionals

A lower semicontinuity theorem for integral functionals is proved underL ₁-strong convergence of the trajectories andL ₁-weak convergence of the control functions. An alternative statement is also proved under pointwise convergence of the trajectories.     

On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.     

On the semicontinuity of Burkill-Cesari integral

We prove a general criterion for the lower semicontinuity of Burkill-Cesari integral in abstract setting. The basic idea, which is inspired by a classical Menger's conjecture, consists on reducing this result to a suitable adaptation of the semicontinuity of the length of a curve. Some noteworthy applications to the Weierstrass functionals of the Calculus of Variations in BV setting are also presented. Dedicated to Professor Calogero Vinti on his 65° birthday     

Solution of a Ball–Murat problem

A necessary and sufficient condition for the W ^{1, p} -quasi-convexity of integrands to imply the lower semicontinuity of the corresponding integral functionals with respect to the weak convergence of sequences in W ^{1, p} is obtained. It is shown that the absence of the Lavrent’ev phenomenon in minimization problems with linear boundary data is sufficient, under a minor technical assumption, for the lower semicontinuity of integral functionals with quasi-convex integrands.     

Lower closure for orientor fields by lower semicontinuity of outer integral functionals

Summary A classical lower semicontinuity result in optimal control theory and the calculus of variations can be extended to outer integral functionals (viz. integral functionals with nonmeasurable integrands). As a consequence, measurability of the Lagrangian does not have to be guaranteed anymore when applying the deparametrization procedure to existence problems in optimal control theory.     

Analysis of nonlinear fractional control systems in Banach spaces

In this paper, we consider the nonlinear control systems of fractional order and its optimal controls in Banach spaces. Using the fractional calculus, Hölder’s inequality, p-mean continuity, weakly singular inequality and Leray-Schauder’s fixed point theorem with compact mapping, the sufficient condition is given for the existence and uniqueness of mild solutions for a broad class of fractional nonlinear infinite dimensional control systems. Utilizing the approximately lower semicontinuity of integral functionals and weakly compactness, we extend the existence result of optimal controls for nonlinear control systems to nonlinear fractional control systems under generally mild conditions. An example is given to illustrate the effectiveness of the results obtained.     

On the existence of optimal solutions in a stochastic control model

An existence result for a stochastic control model with chance constraints, obtained by Christopeit (Ref. 1), is considerably generalized by combining a standard isometry property of Wiener integrals with a well-known lower semicontinuity result for integral functionals.     

Existence Theorems for Periodic Differential Inclusions in IR^N

We study the periodic problem for differential inclusions in R~N.First we look for extremal periodicsolutions.Using techniques from multivalued analysis and a fixed point argument we establish an existencetheorem under some general hypotheses.We also consider the“nonconvex periodic problem”under lowersemicontinuity hypotheses,and the“convex periodic problem”under general upper semicontinuity hypotheseson the multivalued vector field.For both problems,we prove existence theorems under very general hypotheses.Our approach extends existing results in the literature and appear to be the most general results on the nonconvexperiodic problem.     

Approximate selections of almost lower semicontinuous multimaps in C-spaces

We study the relations of almost lower semicontinuity, lower semicontinuity and other generalized lower semicontinuity; then we establish a new approximate selection theorem for almost lower semicontinuous multimaps with the generalized Zima type condition in C$C$-spaces. Our result unify and extend the approximate selection theorems in many published works.     

Existence of optimal controls for nonlinear systems in Banach spaces

Using a recent result of Castaing and Clauzure on the lower semicontinuity of integral functionals, we prove the existence of an optimal control for a broad class of nonlinear infinite-dimensional control systems. An example of a distributed parameter system is worked in detail.This research was supported by NSF Grant No. DMS-84-03135.     

Differential inclusions and Young measures involving prescribed Jacobians

In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (e.g. for incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. This short note, which is based on joint work with K. Koumatos and E. Wiedemann, presents various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension. In particular, we give a characterization theorem for Young measures under this side constraint. This is in the spirit of the celebrated Kinderlehrer–Pedregal Theorem and based on convex integration and “geometry” in matrix space. Finally, applications to approximation in Sobolev spaces and to the failure of lower semicontinuity for certain integral functionals with “realistic” growth conditions are given. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Supremal Representation of L∞ Functionals

We characterize the maps F=F(u,A) defined for u ∈ W^1,∞ and A open, which can be written as supremal functionals of the form F(u,A)=ess sup_{x ∈ A} f(x,u(x),Du(x)).     

Lower semicontinuity and relaxation for integral functionals with <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)- and <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x,u</Emphasis>)-growth

We consider the questions of lower semicontinuity and relaxation for the integral functionals satisfying the p(x)- and p(x, u)-growth conditions. Presently these functionals are actively studied in the theory of elliptic and parabolic problems and in the framework of the calculus of variations. The theory we present rests on the following results: the remarkable result of Kristensen on the characterization of homogeneous p-gradient Young measures by their summability; the earlier result of Zhang on approximating gradient Young measures with compact support; the result of Zhikov on the density in energy of regular functions for integrands with p(x)-growth; on the author’s approach to Young measures as measurable functions with values in a metric space whose metric has integral representation.     

Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands

It is known that sequential weak lower semicontinuity and weak-strong convergence (in the scalar case) properties of integral functionals may be characterized by means of their integrands. In this paper we introduce a Young measure approach obtaining both these results and the characterization for the second property in the vector-valued case. We discuss also motivations for the definition of strict quasiconvexity, and point out that the characterization of the classes of functionals having weak-strong convergence property everywhere is not a trivial problem in the general case.     

A Variational Characterization of the Effective Yield Set for Ionic Polycrystals

The effective yield set of ionic polycrystals is characterized by means of variational principles in $L^\infty $ associated to supremal functionals acting on matrix-valued divergence-free fields.     

Generalized KKM theorems,minimax inequalities,and their applications

This paper extends the well-known KKM theorem and variational inequalities by relaxing the closedness of values of a correspondence and lower semicontinuity of a function. The approach adopted is based on Michael's continuous selection theorem. As applications, we provide theorems for the existence of maximum elements of a binary relation, a price equilibrium, and the complementarity problem. Thus our theorems, which do not require the openness of lower sections of the preference correspondences and the lower semicontinuity of the excess demand functions, generalize many of the existence theorems such as those in Sonnenschein (Ref. 1), Yannelis and Prabhakar (Ref. 2), and Border (Ref. 3).The author is grateful to Professor Franco Giannessi for helpful comments and suggestions.     

Existence of minimizers for a class of quasi-convex functionals with non-standard growth

We study the lower semicontinuity properties of non-autonomous variational integrals whose energy densities satisfy general growth conditions. We apply these results to solve Dirichlet’s boundary value problems for such functionals. Received: June 14, 2000; in final form: November 25, 2000 Published online: December 19, 2001     

Remarks on saddle points in nonconvex sets

Smol'yakov's saddle point theorem is generalized to admissible sets (in the sense of Klee). Moreover, convexity of the involved sets in the theorem can be replaced by acyclicity, and continuity of the involved functions by lower or upper semicontinuity.