Minimizers Of Integral Functionals Research Articles

Фундаментальний аналіз та оптимізація (мінімізація) динамічних навантажень у пружних елементах механізмів підйому вантажу кранів. Частина ІІ

A scientifically substantiated methodology of optimization of motion modes of machines and mechanisms (in particular, city-building cranes) based on minimization of integral functionals, which reflect undesirable properties of mechanical systems, and terminal (initial/final) conditions of motion of the mechanism of lifting cargoes by cranes have a physical meaning and are based on Newton's laws of mechanics, is proposed. The modes of motion of specific mechanical systems with a linear function of the position of elements are optimized on the example of mechanisms of load-lifting city-building cranes. It is shown that the influence of motion modes of drive mechanisms on dynamic loads in elastic elements of the mechanism of lifting mechanism of a city-building crane (ropes) is essential. The parameters of such modes of functioning/movement of such systems, at which exactly dynamic stresses of ropes at the stages of start-up of mechanisms of lifting mechanism of cranes are minimized, are determined. The results obtained in the work can further serve to clarify and improve the existing engineering methods of calculation of load lifting mechanisms of urban cranes in order to minimize the dynamic stresses of their rope systems at the start-up stage, as well as to prevent the occurrence of possible emergency situations. Such an approach can be realized both at the stages of design of the mentioned mechanisms and in the modes of proper operation.

Weak Lower Semicontinuity of Integral Functionals and Applications

Minimization is a reoccurring theme in many mathematical disciplines ranging from pure to applied ones. Of particular importance is the minimization of integral functionals that is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the involved functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century the milestones of the modern theory were set by C.B. Morrey Jr. in 1952 and N.G. Meyers in 1965. We recapitulate the development on this topic from then on. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub)determinants with respect to weak lower semicontinuity. Besides, we emphasize some recent progress in lower semicontinuity of functionals along sequences satisfying differential and algebraic constraints which have applications in elasticity to ensure injectivity and orientation-preservation of deformations. Finally, we outline generalization of these results to more general first-order partial differential operators and make some suggestions for further reading.

The Nikol’Skii Type Regularity of Solutions to Nonlinear Problems in Domains with Hölder Boundary

We study the regularity of solutions to problems of minimization of integral functionals, the Dirichlet and Neumann problems for elliptic operators of order 2m in domains with Holder boundary on a compact Riemannian manifold. We establish interactions between the smoothness of the right-hand side, the regularity of the boundary, and the smoothness of solutions to the problems under consideration. Bibliography: 10 titles.

Vector--valued parabolic $\omega$--minimizers

We are investigating regularity issues for parabolic almost minimizers of integral functionals with polynomial growth structure. In particular, we are interested in functions with values in the target space $\mathbb R^N$ for a dimension $N > 1$ and our aim is to prove Hölder continuity for the spatial gradient of an almost minimizer on an open set of full Lebesgue measure. As classical counter examples show, the regularity results which we achieve are optimal if we are dealing with general almost minimizers and dimensions of the target space strictly greater than one. However, the results are also valid in the case $N=1$, even if they are not optimal in that case. Our contribution can be seen as a natural generalization of the elliptic regularity theory for the gradient of vector--valued almost minimizers, which was developed in the 1980s to the framework of time-dependent, parabolic, problems. The parabolic nature of the problem, in combination with the fact that almost minimizers, in general, do not satisfy any associated parabolic partial differential equation, causes a variety of difficulties in regularity theory, which have to be overcome. The first part of the paper covers integral functionals, which involve merely the spatial gradient of the almost minimizing function. In this case, we obtain a partial Hölder regularity result for the spatial gradient with a Hölder exponent, which can be determined in a quantified way in dependence of the data of the problem. In the second part, we develop the theory for integrands, which involve not only the spatial gradient, but also the almost minimizer itself. In this more general case, the proof becomes substantially more involved, since a priori higher integrability of the spatial gradient has to be used. As a consequence, the explicit control on the Hölder continuity exponent gets lost. This is also in accordance with the elliptic theory where the same phenomenon occurs.

On minimization of integral functionals on the space of measurable functions

On minimization of integral functionals on the space of measurable functions

Remark on the well-posedness of the problem of minimization of integral functionals

In this note, we study the well-posedness of the problem of constrained minimization of an integral functional and show that global well-posedness is equivalent to pointwise well-posedness almost everywhere. Our result extends an earlier one by Zolezzi.

Minimization of integral functionals depending on lipschitz domains

In this paper is solved a minimization problem for what is essentially an integral functional depending on domains which verify an uniform cone property with a fixed parameter θ by extending the techniques land results of O. Caligaris and P. Oliva ‘1’ for convex sets. A Dirichlet condition and an obstacle are considered.

On minimization of integral functionals

On minimization of integral functionals