Primo Brandi

A new graph topology. connections with the compact open topology

Let E be a closed connected subset of and let C be the set of all the closed non-empty subsets of E. Given Ω E С, let GΩ denote the set of all the graphs of continuous functions in . Let . We endow G with a new topology called Τ-topology. It is strictly coarser than Hausdorff metric topology and strictly finer than the topology of uniform convergence of distance functionals on bounded sets of The topological space (G,τ) is homeomorphic to the quotient space with a suitable equivalence relation R. The relationships between τ‐topology and the topologies introduced in GΩ by other Authors are explored.

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.

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.

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.

A hypertopology intended for functional differential equations

Let (X,d) be a locally compact separable metric space, Y a Frechet space and let C be the of all the closed non-empty susbsets of X. Given ω ∈ C let G ω denote the set of all the graphs of continuous functions in C(ω Y). Let G=uω∈cGωWe endow G with a new topology called τ-topology. The topological space (G,τ) is homeomorphic to the quotient space [(C,r)×C(X,Y)]/R with respect to a suitable equivalence relation R. The relationships between τ-topology and the topologies in Cω by other Authors are explored. The results here obtained generlize those got in [12] and find applications in the theory of Ordinary and Partial Differential Equations with hereditary structure.

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-

A Variational Approach to Problems of Plastic Deformation

We present some aspects of a research we have developed in a joint project with L.Cesari and W.H.Yang1 on a variational approach to problems of plastic deformation (3,4,7e,9,16).

Seminormality conditions in the calculus of variations forBV solutions

I(x) <~ ~(x), x e BV(G). In the present paper we introduce a new and more general ,,weak condition F,,, or condition (wF), which subsumes the above mentioned assumptions of the simple and the multiple cases. In fact the couple af seminormality conditions (Q) and (wF) are implied by each set of requirements in [4b] and [4c]. We still prove closure, lower closure and semicontinuity theorems (Sections 1 and 2) which contain the analogous results of both [4b] and [4c]. As discussed in [4abc] and [3], these theorems imply existence theorems for the absolute minimum of the Serrin integral ~.