Moment singularities at angular corners in liquid crystals of nematic type

Summary A linear theory of axial plane reorientation in liquid crystals of nematic type is utilized to examine principal moment singularities which may occur at the vertex of corner boundaries. Reorientation free, moment free, and mixed boundary conditions are considered, and the order of the singularity as a function of the corner angle for these three cases is obtained.

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Zusammenfassung Eine lineare Theorie der Achsenebenen-Neuorientierung an Flüssigkristallen vom nematischen Typ wird verwendet, um die Hauptmoment-Singularität zu untersuchen, die im Scheitel von Begrenzungen durch Ecken auftreten kann. Neuorientierungsfreie, momentenfreie, und gemischte Grenzbedingungen werden behandelt, und die Ordnung der Singularität als Funktion des Eckenwinkels wird für diese drei Fälle bestimmt.

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This work was supported by the United States Energy Research and Development Administration.     

Diseguaglianza di Harnack per equazioni quasilineari ellittiche sopra superficie di curvatura media assegnata

Summary In this paper we prove a Harnack type inequality for non-negative solutions and supersolutions of second order quasilinear elliptic equations on hypersurfaces (inR ⁿ) of L^p prescribed mean curvature, with p>n. In the last section an application to non-parametric surfaces of Lipschitz mean curvature is given.

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Entrata in Redazione il 13 giugno 1977.     

Sul teorema di Riesz-Thorin

Summary We give an extension of the Riesz-Thorin theorem to linear operators which map Morrey spaces in spaces of the same or similar type.

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A Bruno Finzi nel suo 70^mo compleann

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Entrata in Redazione il 10 dicembre 1969.     

Spazi di Sobolev con peso e problemi ellittici in un angolo. (I)

Summary This work, which consists of three parts, is concerned with boundary value problems for linear partial differential equations of elliptic type in a plane sector. They have been investigated within suitable Sobolev weight spaces. In this first part of the work Sobolev weight spaces are studied. Duality, embedding and trace theorems have been proved.

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Lavoro eseguito con contributo del C.N.R. nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni.

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Entrato in Redazione il 28 novembre 1972.     

On the uniqueness of promotion operators on tensor products of type <Emphasis Type="Italic">A</Emphasis> crystals

The affine Dynkin diagram of type A _n ⁽¹⁾ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type A _n crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type A _n crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara’s conjecture that all ‘good’ affine crystals are tensor products of Kirillov-Reshetikhin crystals.     

Su talune disequazioni variazionali ellittiche non lineari del secondo ordine

Summary In this note, making use of a result of J. L. Lions, we examine some non linear elliptic variational inequalities defined on domains which may be unbounded. Such variational inequalities are associated to a uniformely second order elliptic operator. We start with the derivation of an existence theorem (on bounded domains) under non coerciveness assumptions. Next we examine the convergence for the solutions of a collection of variational inequalities. To this purpose we study convergence theorems for variational inequalities associated to operators belonging to a class of abstract mapping of pseudomonotone type between Banach spaces. The solvability of some variational inequalities on unbounded domains then follows directly.

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Entrata in Redazione il 6 aprile 1977.

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Lavoro eseguito nell'ambito del C.N.R., Laboratorio per la Matematica Applicata via L. B. Alberti 4, Genova.     

Teoremi di tipo Liouville e di tipo Phragmen-Lindelöf per le soluzioni di certe equazioni quasiellittiche

Summary Some Liouville type theorems and some Phragmen-Lindel?f type theorems for certain non-homogeneous semielliptic boundary value problems are given.

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A Dario Graffi nel suo 70° compleanno

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Entrata in Redazione il 30 giugno 1975.     

Equazioni ellittiche del secondo ordine di tipo Cordes con termini di ordine inferiore

Summary I study the spectrum of linear second order elliptic partial differentiat equations of Cordes type with lower order terms.

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Lavoro eseguito nell'ambito dell'attività dei gruppi di ricerca matematica del Consiglio Nazionale delle Ricerche.

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Entrata in Redazione il 26 ottobre 1969.     

Fonctions de Morse sur les espaces exotiques

Summary It is shown that on an exotique space V_2n−1 of Kervaire type, we can construct a Morse function having only four critical points, the maximum, the minimum and two points of index n−1 and n.

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Entrata in Redazione il 20 ottobre 1972.     

Spazi di Sobolev con peso e problemi ellittici in un angolo. - II

Summary In part I of this work (see [3]) Sobolev weight spaces related to boundary value problems for linear partial differential equations of elliptic type in a plane sector were discussed. This second part is concerned with such boundary value problems for homogeneous constant coefficient operators. They have been investigated within Sobolev weight spaces included in those previously discussed (see [3]). Existence and uniqueness theorems have been proved

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Entrata in Redazione il 28 dicembre 1972.

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Lavoro eseguito con contributo del C.N.R. nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni.     

Minimum-volume design of elastic trusses with deflection constraints

The paper is concerned with the analysis of minimum-volume design of an elastic truss with deflection constraints. Optimality conditions are obtained by means of the calculus of variations technique utilizing the principle of stationary mutual complementary energy. In comparison with the method based on searching technique in nonlinear programming, it is found that the calculus of variations approach provides us with a better understanding of the general characteristics of the problem. It is shown that for a single loading with a single deflection constraint, the optimum truss is statically determinate. The effects of inequality stress constraints and multiple loadings are examined.

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Zusammenfassung Die Arbeit beschäftigt sich mit der Analyse von Tragwerken minimalen Volumens unter Durchbiegungsbedingungen. Mit Hilfe der Variationsrechnung erhält man Optimalitätsbedingungen, indem man das Prinzip der stationären gegenseitigen komplementären Energie benützt. Im Vergleich zu den Suchverfahren der nichtlinearen Programmierung findet man, dass die Anwendung der Variationsrechnung den allgemeinen Charakter des Problems besser verständlich macht. Es wird gezeigt, dass für eine Einzellast und eine einzige Durchbiegungsbedingung das optimale Tragwerk statisch bestimmt ist.

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Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

On the critical index of Gevrey solvability for some linear partial differential equations

We propose precise estimates of the critical index of Gevrey solvability of some classes of linear partial differential equations with multiple characteristics.

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Sunto Si propongono stime precise dell’indice critico della risolubilità di Gevrey di alcune classi di equazioni lineari differenziali con caratteristiche multiple.

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To the memory of L. Cattabriga

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Research partially supported by a Coordinated Research Project of the University of Cagliari and by INDAM-GNAFA, Italy.     

Problemi quasi-ellittici di trasmissione

Summary We study a transmission problem for quasi-elliptic differential operators (not necessarily of the same type) on a product manifold using pseudo-differential methods.

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Entrata in Redazione il 10 novembre 1973.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

A criterion for non-integrability of Hamiltonian systems with nonhomogeneous potentials

A necessary condition for integrability is presented for two degrees of freedom Hamiltonian systems with nonhomogeneous potentials. This is based on a theorem of Ziglin and gives also an efficient criterion for non-integrability. As an application, it is used to single out integrable cases among the Hénon-Heiles Hamiltonians.

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Résumé On démontre une condition nécessaire pour l'intégrabilité de systèmes hamiltoniens à deux degrés de liberté qui possèdent des potentiels inhomogènes. La démonstration repose sur un théorème de Ziglin et l'on obtient aussi un critère de non-integrabilité. Comme application, on s'en sert pour trouver des cas intégrables dans les hamiltoniens de Hénon-Heiles.

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Dedicated to Prof. Yoshikazu Hirasawa on his 60th birthday

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This research was partially supported by Grant-in-Aid for Scientific Research (No. 60740069), Ministry of Education. Japan.

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Staying at Forschungsinstitut für Mathematik, ETH-Zentrum, Zürich until the end of August, 1987.     

The Painlevé III equation and the Iwasawa decomposition

For the third Painlevé equation an explicit isomorphism between the monodromy data and the data of the approach of Dorfmeister-Pedit-Wu, based on the Iwasawa decomposition of the loop groups, is established. As an application, this provides a simple algebraic way to calculate the monodromy data in terms of the Cauchy data at zero.

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Partially supported by the Sonderforschungsbereich 288

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Partially supported by the National Science Foundation (DMS-9315964)

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This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Un teorema di esistenza per controlli ottimi di sistemi definiti da equazioni integrali di Urysohn

Summary In this paper an existence theorem is proved for optimal control problems described by Urysohn systems, with pointwise constraints on controls and states. An example of application of the existence theorem is given.

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Entrata in Redazione il 15 marzo 1978.

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Lavoro eseguito nell'ambito della mia attività di ricerca come borsista C.N.R., presso l'Università di Genova.     

Multiplicity of solutions for elliptic inequalities in presence of symmetries

Summary In this paper we get a multiplicity result for the solutions of a variational inequality of elliptic type, in presence of symmetries.

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Riassunto Viene fornito un teorema di molteplicità di soluzioni per una disequazione variazionale di tipo ellittico in presenza di simmetrie.

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Work supported by M.P.I. and G.N.A.F.A.     

Über die Realisierbarkeit eines Laser-gepumpten parametrischen Mikrowellen-Verstärkers

Summary The realizability of a parametric microwave amplifier with a laser-pumped electrooptic crystal as a nonlinear element is investigated theoretically for ideal and lossy crystals. It is found that even with the best presently available crystals a 10-db-travelling wave amplifier needs a pump power flux of the order of 10⁷W/cm². Consequentlycw-operation is impossible. Phase matching can be achieved either by a suitable choice of the propagation directions or by increasing the phasevelocity of the signal with a waveguide. Both possibilities are treated. With given parameters of the nonlinear material, the formulas and graphs permit a quick estimation of the required pump power and the proper crystal- and waveguide dimensions. Finally the introduction of a resonator at signal frequency is being considered. However, it is found, that only a small amount of pump power can be saved, as even with low resonatorQ the amplitude fluctuations of the pump deteriorate the gain stability very much. With high resonatorQ a parametric oscillator is obtained.

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Gewidmet Herrn Professor Dr. K. P. Meyer zu seinem 60. Geburtstag     

Su una classe di procedimenti iterativi per la risoluzione di equazioni funzionali e applicazioni

Summary In this paper, we study a class of iterative processes of the type z_n=T_nz_n, which approximate the iterative processes x_n+1=Tx_n, where T and T_n are more general operators than contractions in a metric space.

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Lavoro eseguito in relazione al Contratto N^o 115.3050.0.5189 del Comitato per la Matematica del C.N.R. nel corso dell'anno accademico 1968–69.

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Entrata in Redazione il 25 luglio 1969.     

Über den Satz von Poincaré in der Störungstheorie Briot-Bouquetscher Differentialgleichungen

The well known theorem of H. Poincaré concerning the analytic continuation and representation along a curve of solutions of differential equations depending on a parameter is generalized to solutions of Differential Equations of Briot-Bouquet type depending on a parameter.

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Herrn Professor Dr. N. Hofreiter zum 70. Geburtstag