Anna Salvadori

Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions

SummaryThe authors prove existence theorems for the minimum of multiple integrals of the calculus of variations with constraints on the derivatives in classes of BV possibly discontinuous solutions. To this effect the integrals are written in the form proposed by Serrin. Usual convexity conditions are requested, but no growth condition. Preliminary closure and semicontinuity theorems are proved which are analogous to those previously proved by Cesari in Sobolev classes. Compactness in L1 of classes of BV functions with equibounded total variations is derived from Cafiero-Fleming theorems.

Some variational convergence results with applications to evolution inclusions

We study variational convergence for integral functionals defined on L H ∞ ([0, 1];dt) × y([0,1]; \( \mathbb{Y} \) ) where ℍ is a separable Hilbert space, \( \mathbb{D} \) is a Polish space and y[0,1]; \( \mathbb{D} \) ) is the space of Young measures on [0,1] × \( \mathbb{D} \) , and we investigate its applications to evolution inclusions. We prove the dependence of solutions with respect to the control Young measures and apply it to the study of the value function associated with these control problems. In this framework we then prove that the value function is a viscosity subsolution of the associated HJB equation. Some limiting properties for nonconvex integral functionals in proximal analysis are also investigated.

On convergence in area in the generalized sense

In the present paper, we are concerned with establishing precise connections between convergence in area, convergence in global or separate variations, and convergence in measure of the first partial derivatives of functions f(x, y) on a given rectangle R. Here f(x, u) denotes any summable function in R. The functions f(x, y) of bounded variation in the sense of Cesari, or BVC functions, are those defined geometrically by Cesari [ 1 l] in 1936, and variously designated as functions of generalized bounded variation in the sense of Tonelli, or gBVT. As proved by Krickeberg [ 171 in 1957, the BVC functions are those L-integrable functions f(x,~) whose first-order partial derivatives in the sense of distributions are measures. These BVC functions were used by Smoller and Conway [22] in 1966 to prove existence theorems for weak solutions of shock wave equations or conservative laws in several space variables. The same functions are the object of the theoretical study by Volpert [25 ] on spaces of such elements. The corresponding class of functions of generalized absolute continuity in the sense of Tone& or gACT, can also be defined geometrically. Alternatively, these functions are those L-integrable functions f(x, y) whose partial derivatives in the sense of distributions are L-integrable functions, and thus form the space H’,‘. For L-integrable functions f, or surfaces z =f(x,~), (x, y) E R, the area can be defined as an upper area, or generalized Lebesgue area, Lf, as in Cesari [ 111, where he proved that f has finite upper area Lf if and only iff is BVC. Alternatively, the area can be also delined as a lower area, as in [ 141, and as a Burkill-Cesari integral, af, as in [lo], where we proved that af is finite if and only iff again is BVC. Actually, Lf = af as we proved in [lo]. The present paper takes its motivation from previous work by many

Sull'estensione dell'integrale debole alla Burkill-Cesari ad una misura

We extend the weak Burkill-Cesari integral of a vector valued set function, with values in a Banach space, to a measure, following an idea developed by Cesari ([4], [5]) in Euclidean spaces.

Un teorema di rappresentazione per l'integrale parametrico del Calcolo delle Variazioni alla Weierstrass

SummaryWe prove that the Weierstrass integral over a variety ([5]),in Banach spaces, can be represented as a Bochner integral. This result was given by Cesari ([6])in Euclidean spaces.

A minimax theorem for functions taking values in a Riesz space

On demontre des theoremes de minimax pour des fonctions prenant des valeurs dans un espace de Riesz. On donne une bonne extension de la version en espace de Riesz du theoreme de Hahn-Banach classique

The nonparametric integral of the Calculus of Variations as a Weierstrass integral: Existence and representation☆

Abstract The definition and properties of an abstract and very general nonparametric integral of the Calculus of Variations is presented. In harmony with the Lewy-McShane approach, the nonparametric integral ∝ f, for set functions ϑ taking their values in a Banach space E , is defined in terms of its associated parametric integral. For the latter use is made of the abstract parametric integral proposed by Cesari in R n and then extended to Banach spaces by Breckenridge, Warner, and the authors. A condition (c) is shown to be relevant for the existence of the integral, and is preserved by the nonlinear operation f. Also, for f nonnegative, a Tonelli-type theorem is proved in the sense that the so defined Weierstrass integral ∝ f is always larger than or equal to the corresponding Lebesgue integral, and equality holds if and only if absolute continuity conditions hold. In the proof a suitable martingale is associated and a convergence theorem for martingales is applied. Applications to the calculus of variations will follow.

Some variational convergence results for a class of evolution inclusions of second order using Young measures

This paper has two main parts. In the first part, we discuss the existence and uniqueness of the W E 2,1 -solution u μ,ν of a second order differential equation with two boundary points conditions in a finite dimensional space, governed by controls μ, ν which are measures on a compact metric space. We also discuss the dependence on the controls and the variational properties of the value function V h(t, μ) := supν∈ℜ h(u μ, ν(t)), associated with a bounded lower semicontinuous function h. In the second main part, we discuss the limiting behaviour of a sequence of dynamics governed by second order evolution inclusions with two boundary points conditions. We prove that (up to extracted sequences) the solutions stably converge to a Young measure ν and we show that the limit measure ν satisfies a Fatou-type lemma in Mathematical Economics with variational-type inclusion property.

The nonparametric integral of the Calculus of Variations as a Weierstrass integral. II. Some applications

In [13] we studied the nonparametric integral of the Calculus of Variations as a Weierstrass integral (CV-W integral) in a very general setting. We obtained some existence theorems, a Tonelli-type theorem and consequently a representation theorem in terms of a Lebesgue-Stieltjes integral. We refer to [ 131 for all the notations and definitions. Here we only recall that the CV-W integrals are defined as BurkilllCesari (BC) integrals [ 19,201 of a suitable set function. In this paper we apply our results to the classical integrals of the Calculus of Variations. In particular in [ 131 we studied a weighted CV-W integral (see n. 4a.2). This allows us to introduce here the CV-W integral, depending on the second order differential elements, over a curve of the space. This integral is then compared with the classic one (see Section 2). As it is well-known, the theory of the integral of the Calculus of Variations depending on the second order differential elements was developed by Cinquini [21--231. Successively, Borgogno [9, lo] and Berruti Onesti [ 1, 21 also studied this integral (we mention only some recent papers on this subject). With regard to the applications of our results to the multiple integral of the Calculus of Variations (see Section 3), we improve some well-known results on this subject. In fact we extend to the (extended) Burkill integral some results already given for the strict or intermediate Burkill integral [ 16, 321. Finally we introduce, as a natural application of the weighted CV-W integral already mentioned, the weighted generalized area of a discon-

Inequality systems and minimax results without linear structure

Inequality systems and their connections with minimax results for real- valued functions have been studied in different ways in several papers [ 1, 2,4, 5, 7-9, 133. In particular Pomerol [ 131 takes into consideration the equivalence between the consistency of inequality systems and the resolvability of each inequality of the system; it should be remarked that a minimax theorem can be directly derived from this equivalence. The aim of this paper is to investigate the relationships between the con- sistency of inequality systems and minimax results for functions defined on spaces without linear structure and taking values in a Riesz space; for real-valued functions, minimax results without linear structure go back to Fan [3]. Throughout this paper