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1
Content available remote Remarks on the Bourgain-Brezis-Mironescu approach to Sobolev spaces
100%
Bojarski B.
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2011
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tom 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.
2
Content available remote Multipliers in Sobolev spaces and exact convergence rate estimates for the finite-difference schemes
80%
Jovanović B.
Banach Center Publications
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1994
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tom 29
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nr 1
165-173
EN
In this paper we present some recent results concerning convergence rate estimates for finite-difference schemes approximating boundary-value problems. Special attention is given to the problem of minimal smoothness of coefficients in partial differential equations necessary for obtaining the results.
3
Content available remote Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces
70%
Saksman E. , Soto T.
Analysis and Geometry in Metric Spaces
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2017
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tom 5
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nr 1
98-115
EN
We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.
4
An estimate for the distance of a complex valued Sobolev function defined on the unit disc to the class of holomorphic functions
60%
Fuchs M.
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tom 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.
5
Content available remote Weak solutions to anti-plane boundary value problems in a linear theory of elasticity with microstructure
60%
Shmoylova E. , Potapenko S. , Dorfmann A.
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2007
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tom 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.
6
Content available remote Analysis and numerical approximation of a parabolic-hyperbolic transmission problem
60%
Jovanović B. , Vulkov L.
Open Mathematics
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2012
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tom 10
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nr 1
73-84
EN
In this paper we investigate a mixed parabolic-hyperbolic initial boundary value problem in two disconnected intervals with Robin-Dirichlet conjugation conditions. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate is obtained.
7
Content available Holomorphic Sobolev spaces on the ball
53%
Beatrous F. , Burbea J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction ..............................................................5 0. Preliminaries and notation....................................6 1. Hilbert spaces of holomorphic functions..............11 2. Some estimates ..................................................16 3. The space $L^p_q$............................................22 4. Norm estimates...................................................30 5. Sobolev norms....................................................35 6. Projections in Sobolev spaces............................47 References.............................................................56
8
Content available remote Linear extension operators for restrictions of function spaces to irregular open sets
51%
Rychkov V. S.
Studia Mathematica
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2000
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tom 140
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nr 2
141-162
EN
Let an open set $Ω ⊂ ℝ^n$ satisfy for some 0≤d≤n and ε > 0 the condition: the $d$-Hausdorff content $H_d(Ω∩B) ≥ ε|B|^{d/n}$ for any ball B centered in Ω of volume |B|≤1. Let $H_p^s$ denote the Bessel potential space on $ℝ^n$ 1 < p < ∞,s > 0, and let $H_p^s[Ω]$ be the linear space of restrictions of elements of $H_p^s$ to Ω endowed with the quotient space norm. We find sufficient conditions for the existence of a linear extension operator for $H_p^s[Ω]$, i.e., a bounded linear operator $H_p^s[Ω]→H_p^s$ such that $ext⨍|_Ω}=⨍$ for all ⨍. The main result is that such an operator exists if (i) d > n-1 and s > (n-d)/min(p,2), or (ii) d≤n-1 and s-[s] > (n-d)/min(p,2). It is an open problem whether these assumptions are sharp.
9
Content available remote On a nonlinear Kirchhoff-Carrier wave equation in the unit membrane: the quadratic convergence and asymptotic expansion of solutions
51%
Long N. , Ngoc L.
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2007
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tom Vol. 40, nr 2
365-392
EN
In this paper we consider the initial-boundary value problem for the nonlinear wave equation.
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