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1
Content available Positive solutions for nonparametric anisotropic singular solutions
Papageorgiou Nikolaos S., Rădulescu Vicenţiu D., Sun Xueying
Opuscula Mathematica
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2024
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Vol. 44, no. 3
409--423
EN
We consider an elliptic equation driven by a nonlinear, nonhomogeneous differential operator with nonstandard growth. The reaction has the combined effects of a singular term and of a “superlinear” perturbation. There is no parameter in the problem. Using variational tools and truncation and comparison techniques, we show the existence of at least two positive smooth solutions.
2
Content available Existence and asymptotic stability for generalized elasticity equation with variable exponent
Dilmi Mohamed, Otmani Sadok
Opuscula Mathematica
|
2023
|
Vol. 43, no. 3
409--428
EN
In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor σp(·) has the form σp(·)(u) =(2μ + |d(u)|p(·)−2) d(u) + λTr (d(u)) I3, where u is the displacement field, μ, λ are the given coefficients d(·) and I3 are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions.
3
Content available On the Lebesgue and Sobolev spaces on a time-scale
Skrzypek Ewa, Szymańska-Dębowska Katarzyna
Opuscula Mathematica
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2019
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Vol. 39, no. 5
705--731
EN
We consider the generalized Lebesgue and Sobolev spaces on a bounded time-scale. We study the standard properties of these spaces and compare them to the classical known results for the Lebesgue and Sobolev spaces on a bounded interval. These results provide the necessary framework for the study of boundary value problems on bounded time-scales.
4
Content available On the numerical solution of the initial-boundary value problem with neumann condition for the wave equation by the use of the laguerre transform and boundary elements method
Litynskyy S., Muzychuk Y., Muzychuk A.
Acta Mechanica et Automatica
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2016
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Vol. 10, no. 4
285--290
EN
We consider a numerical solution of the initial-boundary value problem for the homogeneous wave equation with the Neumann condition using the retarded double layer potential. For solving an equivalent time-dependent integral equation we combine the Laguerre transform (LT) in the time domain with the boundary elements method. After LT we obtain a sequence of boundary integral equations with the same integral operator and functions in right-hand side which are determined recurrently. An error analysis for the numerical solution in accordance with the parameter of boundary discretization is performed. The proposed approach is demonstrated on the numerical solution of the model problem in unbounded three-dimensional spatial domain.
5
Content available On a boundary value problem for the system of partial differential equations describing non-simple thermoelasticity
Kołakowski H., Łazuka J.
Biuletyn Wojskowej Akademii Technicznej
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2014
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Vol. 63, nr 2
171-176
PL
W pracy rozważamy zagadnienie Dirichleta dla równań teorii termosprężystości materiałów złożonych. Pokazaliśmy, że zagadnienie to generuje operator fredholmowski działający pomiędzy odpowiednimi przestrzeniami Sobolewa.
EN
In this paper we study the Dirichlet problem for the system of equations describing non-simple thermoelasticity. Using the general theory of the elliptic problem we show that this problem is elliptic one.
6
Content available remote Characterizations of some function spaces in terms of Haar wavelets
Triebel H.
Commentationes Mathematicae
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2013
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Vol. 53, No. 2
35--53
EN
Some spaces Asp,q(Rn) with A = {B, F}, s ϵ R, 0 < p, q ≤ ∞, covering Besov spaces, Hölder-Zygmund spaces and Sobolev spaces, admit characterizations in terms of Haar bases. It is the main aim of this paper to extend this observation to corresponding Morreyfied spaces Lr Asp,q(Rn). As a by-product we obtain Littlewood-Paley theorems for (homogeneous and inhomogeneous) Morrey spaces Lrp(Rn), Lrp(Rn) and, in particular, L°rp(Rn).
7
Content available remote 1D Dirac Operators with Special Periodic Potentials
Djakov P., Mityagin B.
Bulletin of the Polish Academy of Sciences. Mathematics
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2012
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Vol. 60, no 1
59--75
EN
For one-dimensional Dirac operators of the form [formula] we single out and study a class X of π-periodic potentials v whose smoothness is determined only by the rate of decay of the related spectral gaps γn=|λ+n−λ−n|, where λ±n are the eigenvalues of L=L(v) considered on [0,π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions.
8
An estimate for the distance of a complex valued Sobolev function defined on the unit disc to the class of holomorphic functions
Fuchs M.
Journal of Applied Analysis
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2011
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Vol. 17, nr 1
131-135
EN
Let f : B →C denote a Sobolev function of class W1p defined on the unit disc. We show that the distance of f to the class of all holomorphic functions measured in the norm of the space W1p(B;C) is bounded by the Lp-norm of theWirtinger derivative ∂-zf. As a consequence we obtain a Korn type inequality for vector fields B →R2.
9
Content available remote Remarks on the Bourgain-Brezis-Mironescu approach to Sobolev spaces
Bojarski B.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
Vol. 59, no 1
65-75
EN
For a function ƒ ∈ L[wzór] (Rn) the notion of p-mean variation of order 1, V[wzór](ƒ, Rn) is defined. It generalizes the concept of F. Riesz variation of functions on the real line R1 to Rn, n > 1. The characterisation of the Sobolev space W1,p(Rn) in terms of V[wzór](ƒ, Rn) is directly related to the characterisation of W1,p(Rn) by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.
10
Content available remote On a nonlinear Kirchhoff-Carrier wave equation in the unit membrane: the quadratic convergence and asymptotic expansion of solutions
Long N., Ngoc L.
Demonstratio Mathematica
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2007
|
Vol. 40, nr 2
365-392
EN
In this paper we consider the initial-boundary value problem for the nonlinear wave equation.
11
Content available remote Weak solutions to anti-plane boundary value problems in a linear theory of elasticity with microstructure
Shmoylova E., Potapenko S., Dorfmann A.
Archives of Mechanics
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2007
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Vol. 59, nr 6
519-539
EN
In this paper we formulate the interior and exterior Dirichlet and Neumann boundary value problems of anti-plane rnicropolar elasticity in a weak (Sobolev) space setting, we show that these problems are well-posed and the corresponding weak solutions depend continuously on the data. We show that the problem of torsion of a rnicropolar beam of (non-smooth) arbitrary cross-section can be reduced to an interior Neumann boundary value problem in antiplane micropolar elasticity and consider an example which demonstrates the significance of material microstructure.
12
Content available remote On constants in the pointwise one-dimensional multiplicative inequalities
Kałamajska A.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2002
|
[Z] 42(1)
37-51
EN
We give the bounds on constants in the one-dimensional pointwise multiplicative inequalities [formula] where Mf(x) is the Hardy-Littlewood maximal function of f. Our constants are optimal in the case k=1, m=2.
13
Content available remote On a Compact Extension
Hofmanová P.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
1999
|
Vol. 6
29--31
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