Necessary condition for eigensolutions of the 1-Laplace operator by means of inner variations

2012 ◽  
Vol 356 (1) ◽  
pp. 147-177 ◽  
Author(s):  
Zoja Milbers ◽  
Friedemann Schuricht
2007 ◽  
Vol 09 (04) ◽  
pp. 515-543 ◽  
Author(s):  
BERND KAWOHL ◽  
FRIEDEMANN SCHURICHT

We consider a number of problems that are associated with the 1-Laplace operator Div (Du/|Du|), the formal limit of the p-Laplace operator for p → 1, by investigating the underlying variational problem. Since corresponding solutions typically belong to BV and not to [Formula: see text], we have to study minimizers of functionals containing the total variation. In particular we look for constrained minimizers subject to a prescribed [Formula: see text]-norm which can be considered as an eigenvalue problem for the 1-Laplace operator. These variational problems are neither smooth nor convex. We discuss the meaning of Dirichlet boundary conditions and prove existence of minimizers. The lack of smoothness, both of the functional to be minimized and the side constraint, requires special care in the derivation of the associated Euler–Lagrange equation as necessary condition for minimizers. Here the degenerate expression Du/|Du| has to be replaced by a suitable vector field [Formula: see text] to give meaning to the highly singular 1-Laplace operator. For minimizers of a large class of problems containing the eigenvalue problem, we obtain the surprising and remarkable fact that in general infinitely many Euler–Lagrange equations have to be satisfied.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sharief Deshmukh ◽  
Olga Belova ◽  
Nasser Bin Turki ◽  
Gabriel-Eduard Vîlcu

AbstractWe study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ u = − φ ( N ) . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ S 2 n + 1 . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ R i c ( u , u ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ S 2 n + 1 .


1997 ◽  
Vol 161 ◽  
pp. 267-282 ◽  
Author(s):  
Thierry Montmerle

AbstractFor life to develop, planets are a necessary condition. Likewise, for planets to form, stars must be surrounded by circumstellar disks, at least some time during their pre-main sequence evolution. Much progress has been made recently in the study of young solar-like stars. In the optical domain, these stars are known as «T Tauri stars». A significant number show IR excess, and other phenomena indirectly suggesting the presence of circumstellar disks. The current wisdom is that there is an evolutionary sequence from protostars to T Tauri stars. This sequence is characterized by the initial presence of disks, with lifetimes ~ 1-10 Myr after the intial collapse of a dense envelope having given birth to a star. While they are present, about 30% of the disks have masses larger than the minimum solar nebula. Their disappearance may correspond to the growth of dust grains, followed by planetesimal and planet formation, but this is not yet demonstrated.


Author(s):  
G.D. Danilatos

The environmental scanning electron microscope (ESEM) has evolved as the natural extension of the scanning electron microscope (SEM), both historically and technologically. ESEM allows the introduction of a gaseous environment in the specimen chamber, whereas SEM operates in vacuum. One of the detection systems in ESEM, namely, the gaseous detection device (GDD) is based on the presence of gas as a detection medium. This might be interpreted as a necessary condition for the ESEM to remain operational and, hence, one might have to change instruments for operation at low or high vacuum. Initially, we may maintain the presence of a conventional secondary electron (E-T) detector in a "stand-by" position to switch on when the vacuum becomes satisfactory for its operation. However, the "rough" or "low vacuum" range of pressure may still be considered as inaccessible by both the GDD and the E-T detector, because the former has presumably very small gain and the latter still breaks down.


Author(s):  
E. N. Shiryaeva ◽  
M. A. Polyakov ◽  
D. V. Terent'ev

Complexity of modern metallurgical plants, presence of great number of horizontal and vertical interactions between their various structural subdivisions makes it necessary to apply a systems analysis to elaborate effective measures for stable development of a plant operation. Among such measures, digitalization of a plant is widespread at present. To implement the digitalization it is necessary to have clear vision about links at all the levels of the technological system of a plant. A terminology quoted, accepted in the existing regulatory documents for defining of conceptions, comprising the technological system. It was shown, that the following four hierarchical levels of technological systems are distinguished: technological systems of operations, technological systems of processes, technological systems of production subdivisions and technological systems of plants. A hierarchical scheme of technological systems of hot-rolled sheet production at an integrated steel plant presented. Existing horizontal and vertical links between the basic plant’s shops shown. Peculiarities of flows of material, energy and information at the operation “rolling” of the technological system “hot rolling of a steel sheet” considered. As a technical system of the technological process of the hot rolling, the hot rolling mill was chosen. A structural diagram of the hot rolling mill was elaborated, the mill being consisted of reheating furnaces, roughing and finishing stand groups, with an intermediate roll-table between them, and down-coilers section. Since the rolling stands are the basic structural elements of the hot rolling mill, structural diagrams of a roughing and a finishing stands were elaborated. Results of the systems analysis of the technological and technical systems, hierarchically linked in the process of steel sheet hot rolling, can be applied for perfection of organization structure of the whole plant, as well as for elaboration mathematical models of a system separate elements functioning, which is a necessary condition for a plant digitalization.


Author(s):  
B. A. Sivak ◽  
A. V. Protasov ◽  
L. A. Smirnov

The removal of slag from melt surface of hot metal and steel ladles is a necessary condition to provide a deep desulphurization and dephosphorization of hot metal and steel in the process of their processing. A review of methods of slag skimming presented, mainly based on slag mechanical shoveling and its removal out of ladles by vacuum sucking. It was shown, that manipulators design for the slag skimming working instrument moving depends on the production scale, mass of the processed heats, amount and properties of the slag to be removed, production process intensity and ecological requirements. Peculiarities of designs and technical parameters of machines for slag skimming presented, designed by Irkutsk plant of heavy machinery, Scientific and Production Enterprise n.a. M.I. Platov, VNIIMETMASH, Kuznetsk and Novolipetsk steel plants. Technological methods of control of slag composition and physical properties considered, first of all of viscosity and fluidity, which have significant effect on selection of a method of slag skimming. Advantages and drawbacks of actions, aimed at more complete slag removal from metal surface by a scraper noted including bath blowing off by an inert gas, liquid slag tapping into am intermediate settling tank following its removal into a slag bowl. In case of satisfactory fluidity it is possible to slag removal by vacuum sucking, which at the same time promoted the melt degassing. Work done in this area abroad noted. Methods of vacuum slag removal developed in the USA and Japan described.


2007 ◽  
pp. 5-27 ◽  
Author(s):  
J. Searle

The author claims that an institution is any collectively accepted system of rules (procedures, practices) that enable us to create institutional facts. These rules typically have the form of X counts as Y in C, where an object, person, or state of affairs X is assigned a special status, the Y status, such that the new status enables the person or object to perform functions that it could not perform solely in virtue of its physical structure, but requires as a necessary condition the assignment of the status. The creation of an institutional fact is, thus, the collective assignment of a status function. The typical point of the creation of institutional facts by assigning status functions is to create deontic powers. So typically when we assign a status function Y to some object or person X we have created a situation in which we accept that a person S who stands in the appropriate relation to X is such that (S has power (S does A)). The whole analysis then gives us a systematic set of relationships between collective intentionality, the assignment of function, the assignment of status functions, constitutive rules, institutional facts, and deontic powers.


2015 ◽  
pp. 87-108 ◽  
Author(s):  
A. Knobel

The paper is devoted to the analysis of development prospects and problems of Eurasian Economic Union (EAEU) of Russia, Kazakhstan, Belarus and Armenia. It considers integration problems inside EAEU, interactions of EAEU with other CIS countries and with countries from the rest of the world. The paper shows that the major integration challenge inside EAEU is the domination of the redistributive motive over the creative one. It estimates the value of the oil and gas transfer from Russia to other EAEU members and the influence of the Russian tax maneuver on this transfer. The paper shows the need in redistribution mechanism inside EAEU as a necessary condition for getting the potential positive economic effects of free trade with other countries. It also assesses the risks for EAEU due to Russian embargo for food imports from countries of the sanctions list and possible application of tariff in the trade with Ukraine.


1970 ◽  
Vol 8 (1) ◽  
pp. 165-172
Author(s):  
Оксана Воронкевич

У   статті   актуалізовано   проблему   поширеності   шкільного   насильства   у   середовищі   учнів  початкових класів. Особлива увага звертається на необхідність діалогічної взаємодії учасників освітнього  процесу як необхідної умови попередження насилля у школі.  Опираючись на результати власного дослідницького пошуку, автор пропонує варіант програми  психологічної профілактики шкільного насильства вчителів стосовно учнів. Дана програма спрямована на  формування  у  педагогів  навичок  глибинного  самопізнання  й  пізнання  дітей,  апробування  нових  форм  поведінки та базується на ідеї діалогізації педагогічної взаємодії, оскільки важливо налагодити суб’єкт- суб’єктну взаємодію учня та вчителя й не використовувати монологічну модель спілкування. Наголошено,  що педагоги повинні стимулювати будь-які прояви суб’єктної активності дітей, що сприяють виробленню  у них адекватної оцінки себе та свого оточення, розвитку здатності до самовизначення. Відзначено, що  для   діалогічного   освітнього   середовища   характерними   є   такі   властивості,   як   різноманітність,  динамічність, напруженість, достатність, кожна з яких сприяє високій ефективності освітньої взаємодії,  здійснює істотний вплив на розвиток особистості. Під час занять використано різні тренінгові методи:  рольові ігри, міні-лекції, мозковий штурм, обговорення в загальному колі тощо. Разом із тим поширено  інформацію з актуальних для педагогів питань спілкування з дитиною без агресії, злості та конфлікту.  Представлено  результати  успішної  апробації  програми  психологічної  профілактики  шкільного  насильства  з  боку  вчителів,  що проявилися в розумінні важливості толерантного ставлення до учнів,  набутті практичних умінь відчувати психологічний стан іншої людини та адекватно реагувати на нього,  виявляти доброзичливість, прихильність до школярів та надавати їм необхідну допомогу.  The article actualizes the problem of the prevalence of school violence among elementary school pupils.  Special attention is drawn to the need for dialogical interaction of participants in the educational process as a  necessary condition for preventing violence in school.  Relying on the results of his own research, the author suggests a variant of school violence psychological  prevention program of teachers in relation to pupils. This program is aimed at educating the students the skills of  deep self-cognition and cognition of children, testing new forms of behavior and is based on the idea of pedagogical  interaction dialogization, since it is important to establish subject-subject interaction between the pupil and the  teacher and not use the monologue model of communication. It is highlighted that teachers should stimulate any  manifestation of children's subject activity, which helps to develop an adequate assessment of themselves and their  environment, development of self-determination ability. It is noted that a dialogical educational environment is  characterized by such attributes as diversity, dynamism, intensity, sufficiency, each of them contributes to the high  effectiveness of educational interaction, have a significant impact on the development of personality. During the  classes various training methods were used: role-playing games, mini-lectures, brainstorming, discussions in the  general circle, etc. At the same time, information on relevant for teachers issues about communicating with a child  without aggression, anger and conflict is propagated.  Was presented results of successful approbation of the school violence psychological prevention program  from teacher’s part, manifested in the understanding importance of the tolerant attitude towards pupils, acquiring  practical skills of feeling the another’s person psychological state and react adequately to it, showing benevolence,  adherence to schoolchildren and providing them the necessary assistance. 


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