Problemi discontinui e di trasmissione per due equazioni ellittiche di ordine diverso a coefficienti variabili relativi ad angoli consecutivi

Summary In this paper we continue our investigations (see[5]) devoted to transmission problems in consecutive angles, for elliptic operators. In[5] the above mentioned boundary value problems in the case of homogeneous constant coefficient operators and within the Sobolev weight spaces W _s1,s2 ^r were investigated. This work is concerned with the problem for general operators, within the Sobolev weight spaces W _s1,s2 ^r and W _s1,s2 ^*r . Existence theorems and stability theorems for nullity, deficiency, index have been proved.

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Entrata in Redazione il 28 marzo 1975.     

On the Sobolev problem in the complete scale of Banach spaces

In a bounded domainG with boundary ∂G that consists of components of different dimensions, we consider an elliptic boundary-value problem in complete scales of Banach spaces. The orders of boundary expressions are arbitrary; they are pseudodifferential along ∂G. We prove the theorem on a complete set of isomorphisms and generalize its application. The results obtained are true for elliptic Sobolev problems with a parameter and parabolic Sobolev problems as well as for systems with the Douglis-Nirenberg structure. Chernigov Pedagogical Institute, Chernigov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1181–1192, September, 1999.     

Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

 Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]). (Received 27 October 1998; in revised form 17 May 1999)     

Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients

Summary I give a sufficient condition in order that a Dirichlet problem is solvable in H ² (Ω) for a class of linear second order elliptic partial differential equations. Such a class includes some particular cases for which the result is known.

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Sunto Si prova una condizione sufficiente affinchè un problema di Dirichlet sia risolubile in H ² (Ω) per una classe di equazioni differenziali alle derivate parziali lineari ellittiche del secondo ordine. Tale classe comprende alcuni casi particolari per i quali il risultato è noto.

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The present work was written while the author was a member of the ? Centro di Matematica e Fisica Teorica del C.N.R. ? at the University of Genova, directed by professorJ. Cecconi.

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Entrata in Redazione il 25 febbraio 1971.     

Esistenza,unicità e dipendenza continua per una classe di problemi ai limiti non lineari

Summary The following abstract boundary value problem is considered: Lx=Hx, Qx=0, where L, H, Q denote operators, L linear, mapping normed spaces into normed spaces. Existence and uniqueness criteria are given and the dependence of the solution on L, H, Q is studied.

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Lavoro eseguito nell'ambito dell'attività dei Gruppi di ricerca matematica del C.N.R. I numeri 1–5 sono dovuti aG. Pulvirenti e i numeri 6–10 aG. Santagati.     

The generalized solution of ill-posed boundary problem

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk,p0(Ω,∑). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.     

Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

 Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]).     

Compacité faible dans l'espaceL E 1 et dans l'espace des multifonctions intégralement bornées,et minimisation (*).

Summary Some compactness results in the space of Bochner integrable functions in Banach spaces and in the space of integrably bounded multifunction with non empty convex weakly compact values are presented. Applications to minimization problems are given.

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Pour terminer, nous remercions M.Valadier qui a lu attentivement ce papier.     

Su alcuni problemi fortemente ben posti di controllo ottimo

Summary Let P_n be a sequence of optimal control problems with fixed end times (described by ordinary differential equations, with pointwise and norm (of controls) constraints, and with general functional). Let P₀ be an ? unperturbed ? problem (of the same type). In this paper theorems are obtained about the strong convergence (in some L^p) of optimal controls of P_n to some optimal control of P₀, about the uniform convergence x_n → x₀ of states, and the property thatmin P_n min P₀. As corollaries, convergence theorems for some calculus of variations problems can be derived. Weak convergence theorems of optimal controls of P_n to an optimal one of P₀ were considered in[7]. A general abstract theorem about strong convergence of minimum points, generalizing a result in[5], is proved.

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Indirizzo dell'autore: Istituto di Matematica, via L. B. Alberti 4 - 16132 Genova. Lavoro eseguito nell'ambito del Centro di Matematica e di Fisica Teorica del C.N.R. presso l'Università di Genova.

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Entrata in Redazione il 4 maggio 1972.     

Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions

We analyze a class of quasilinear elliptic problems involving a p(·)-Laplace-type operator on a bounded domain W ì \mathbb R^N{\Omega\subset{\mathbb R}^N}, N ≥ 2, and we deal with nonlinear conditions on the boundary. Working on the variable exponent Lebesgue–Sobolev spaces, we follow the steps described by the “fountain theorem” and we establish the existence of a sequence of weak solutions.     

Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H^∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L^p(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω.     

Teoremi merceriani relativi a trasformazioni regolari non negative e anche più generali

Sunto Ad una successione (complessa) |s_n| associamo la successione trasformata |t_n| mediante una matrice di sommazione regolare C; sono noti teoremi merceriani (per es.G. H. Hardy, E. R. Love) che dalla relazione s_n — qt_n → (1 — q)l deducono s_n → l, sotto la condizione che il moltiplicatore q sia interno a determinati campi In questa NOta si considera C regolare non negativa e, mediante l'introduzione di un' opportuna ? funzione di aderenza di un insieme in un altro ? e l'applicazione del classico teorema del ? nocciolo ? diK. Knopp, si determinano campi più ampi di quello circolare stabilito daE. R. Love; si estende il risultato anche a matrici C regolari più generali mediante teoremi più recenti diR. P. Agnew eA. Robinson; inoltre si perfeziona il classico teorema diJ. Mercer-G. H. Hardy.

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Summary Let |t_n| be trasformed, by a regular summation matrix C, of a (complex) sequence |s_n|. Mercerian theorems (e. g. G. H. Hardy, E. R. Love theorems) deduce s_n → l from s_n — qt_n → (1 — q)l if q lies in a suitable domain. This paper is concerned with regular non-negative C. A suitable ? adherence-function of a set into another ? is introduced and theKnopp's ? Core theorem ? leads to wider domains thanE. R. Love circle. The result is extended to more general regular matrices C by means ofR. P. Agnew andA. Robinson theorems. The classicalJ. Mercer-G. H. Hardy theorem is also improved.

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A Giovanni Sansone nel suo 70^mo compleanno.

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Il presente lavoro è stato oggetto di una comunicazione al VI Congresso Nazionale della Unione Matematica Italiana (Napoli, 11–16 settembre 1959).     

Besov regularity for second order elliptic boundary value problems with variable coefficients

This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝ ^d . The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions. This work has been supported by the Deutsche Forschungsgemeinschaft (Da 360/1-1)     

Proprietà spettrali di una classe di operatori quasi ellittici

Riassunto In questo lavoro ci occupiamo di problemi al contorno per una classe di operatori lineari quasi ellittici del tipoA+σ, dove σ è una perturbazione assegnata; proviamo che l'operatore associato al problema è di Fredholm con indice zero e studiamo le proprietà spettrali dell'operatore associato al problema con condizioni al contorno omogenee.

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Summary In this paper we study boundary value problems for linear quasi elliptic (A+σ)-type operators, where σ is an assigned perturbation; we prove that the related operator is a Fredholm operator whose index is zero. Then we study spectral properties of the operator related to the problem with homogeneous boundary conditions.

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Lavoro eseguito nell'Istituto di Scienze dell'Informazione dell'Università di Salerno, nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni del C.N.R.     

Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [10] on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in L ²-norm as long as the prescribed angular velocity α(t) of the boundary has bounded total variation. Here we establish convergence in stronger L ² and L ^p -Sobolev spaces, allow for more singular angular velocities α, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently. Supported in part by NSF grant DMS-0456861.     

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.     

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 elliptic systems in Hörmander spaces

We study a linear system of pseudodifferential equations uniformly elliptic in Petrovskii’s sense in the Hilbert scale of H?rmander functional spaces defined in ℝ _n . An a priori estimate is proved for the solution of the system and its interior smoothness in this scale of spaces is investigated. As an application, we establish a sufficient condition for the existence of continuous bounded derivatives of the solution.     

On saturation effects in the Neumann boundary control of elliptic optimal control problems

A Neumann boundary control problem for a linear-quadratic elliptic optimal control problem in a polygonal domain is investigated. The main goal is to show an optimal approximation order for discretized problems after a postprocessing process. It turns out that two saturation processes occur: The regularity of the boundary data of the adjoint is limited if the largest angle of the polygon is at least 2π/3. Moreover, piecewise linear finite elements cannot guarantee the optimal order, if the largest angle of the polygon is greater than π/2. We will derive error estimates of order h ^α with α∈[1,2] depending on the largest angle and properties of the finite elements. Finally, numerical test illustrates the theoretical results.     

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.