Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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An order drop theorem
Get PDFA drop theorem on ordered metric spaces is established from the (pre) order version of Ekeland’s variational principle in Turinici [An St UAIC Ia (Math), 36 (1990), 329-352]. The logical equivalence between these results is also discussed
The null space dimension and the existence of the interpolating spline-function in Banach space
No full textUsing the results in papers [2] and [3] in this paper we prove the existence of the interpolating spline-function by the null space dimension of operators A and T
Diffraction of electromagnetic wave on the system of metallic strips in the stratified medium
Get PDFIn this paper, we study diffraction of electromagnetic wave on the system of metallic strips in the stratified medium. The integral equation which represent this problem is solved by Galerkin’s method. To this equation the plane diffraction problem for TE-polarizable electromagnetic wave on the system of metallic strips in the stratified medium was reduced
Biharmonic Green functions
Get PDFThe harmonic Green and Neumann function and a particular Robin function are used to construct bi-harmonic Green, Neumann and particular Robin functions. Moreover hybrid bi-harmonic Green functions are given. They all are constructed via a convolution of the mentioned harmonic particular fundamental solutions. In case of the unit disc they are explicitly expressed. Besides these 9 bi-harmonic Green functions there is another bi-harmonic Green function in explicit form for the unit disc not defined by convolution. Related boundary value problems are not all well posed. In case they are, the unique solutions are given. For the other cases solvability conditions are determined and the unique solutions found. There are all together 12 Dirichlet kind boundary value problems for the inhomogeneous bi-harmonic equation treated. The investigation is restricted to the two dimensional case and complex notation is used
COMPARISON RESULTS FOR SOLUTIONS OF PARABOLIC EQUATIONS WITH A SINGULAR POTENTIAL
Get PDFWe consider the solution u of the Cauchy-Dirichlet problem for a class of linear parabolic equations in which the coefficient of the zero order term could have a singularity at the origin of the type 1/|x|^2 . We prove that u can be compared “in the sense of rearrangements” with the solution v of a problem whose data are radially symmetric with respect to the space variable.We consider the solution u of the Cauchy-Dirichlet problem for a classof linear parabolic equations in which the coefficient of the zero order term could have a singularity at the origin of the type 1/|x|^2. We prove that u can be compared “in the sense of rearrangements” with the solution v of a problem whose data are radially symmetric with respect to the space variable
STABILITY AND FREE ENERGIES IN LINEAR VISCO-ELASTICITY
Get PDFDedicated to the memory of Gaetano FicheraIn this paper we show that several free energies can be related to a material with fading memory with different domains of definition and topologies. As Fichera has proved, the study of stability can be affected by the space and then by the topology which we chose, while the material must be independent by this one.In the last part of the work by the semi-group theory, we use the new notion of minimum state to study the asymptotic behavior of the differential system.Dedicated to the memory of Gaetano FicheraIn this paper we show that several free energies can be related to a material with fading memory with different domains of definition and topologies. As Fichera has proved, the study of stability can be affected by the space and then by the topology which we chose, while the material must be independent by this one.In the last part of the work by the semi-group theory, we use the newnotion of minimum state to study the asymptotic behavior of the differential system
On the boundary behavior of the holomorphic sectional curvature of the Bergman metric
Get PDFWe obtain a conceptually new differential geometric proof of P. F. Klembeck’s result (cf. [9]) that the holomorphic sectional curvature k g (z ) of the Bergman metric of a strictly pseudoconvex domain Ω ⊂ ℂ n approaches −4/(n + 1) (the constant sectional curvature of the Bergman metric of the unit ball) as z → ∂ Ω
COMPLETENESS THEOREMS: FICHERA’S FUNDAMENTAL RESULTS AND SOME NEW CONTRIBUTIONS
Get PDFAfter recalling Fichera’s fundamental results in the study of the problem of the completeness of particular solutions of a partial differential equation, we give some new completeness theorem. They concern the Dirichlet problem for a general elliptic operator of higher order with real constant coefficients in any number of variables.After recalling Fichera’s fundamental results in the study of the problem of the completeness of particular solutions of a partial differential equation, we give some new completeness theorem. They concern theDirichlet problem for a general elliptic operator of higher order with realconstant coefficients in any number of variables
EIGENVECTORS AND FIXED POINTS OF NON-LINEAR OPERATORS
Get PDFLet X be a real infinite-dimensional Banach space and ψ a measure of noncompactness on X. Let Ω be a bounded open subset of X and A : Ω → X a ψ-condensing operator, which has no fixed points on ∂Ω.Then the fixed point index, ind(A,Ω), of A on Ω is defined (see, for example, ([1] and [18]). In particular, if A is a compact operator ind(A,Ω) agrees with the classical Leray-Schauder degree of I −A on Ω relative to the point 0, deg(I −A,Ω,0). The main aim of this note is to investigate boundary conditions, under which the fixed point index of strict- ψ-contractive or ψ-condensing operators A : Ω → X is equal to zero. Correspondingly, results on eigenvectors and nonzero fixed points of k-ψ-contractive and ψ-condensing operators are obtained. In particular we generalize the Birkhoff-Kellog theorem [4] and Guo’s domain compression and expansion theorem [17]. The note is based mainly on the results contained in [7] and [8].Let X be a real infinite-dimensional Banach space and ψ a measureof noncompactness on X. Let Ω be a bounded open subset of X andA : Ω → X a ψ-condensing operator, which has no fixed points on ∂Ω.Then the fixed point index, ind(A,Ω), of A on Ω is defined (see, for example, ([1] and [18]). In particular, if A is a compact operator ind(A,Ω) agrees with the classical Leray-Schauder degree of I −A on Ω relative to the point 0, deg(I −A,Ω,0). The main aim of this note is to investigate boundary conditions, under which the fixed point index of strict- ψ-contractive or ψ-condensing operators A : Ω → X is equal to zero. Correspondingly, results on eigenvectors and nonzero fixed points of k-ψ-contractive and ψ-condensing operators are obtained. In particular we generalize the Birkhoff-Kellog theorem [4] and Guo’s domain compression and expansion theorem [17]. The note is based mainly on the results contained in [7] and [8]
FRACTAL AND EUCLIDEAN INTERACTION IN SOME TRANSMISSION PROBLEMS
Get PDFIn this talk some model examples of second order elliptic transmission problems with highly conductive layers will be described. Regularity and numerical results for solutions of transmission problems across fractal layers imbedded in Euclidean domains will be presented in the aim of better understanding the analytical problems which arise when fractal and Euclidean structures mutually interact.In this talk some model examples of second order elliptic transmissionproblems with highly conductive layers will be described. Regularity andnumerical results for solutions of transmission problems across fractallayers imbedded in Euclidean domains will be presented in the aim ofbetter understanding the analytical problems which arise when fractal and Euclidean structures mutually interact