scholarly journals On density of compactly supported smooth functions in fractional Sobolev spaces

2021 ◽  
Author(s):  
Bartłomiej Dyda ◽  
Michał Kijaczko
Keyword(s):  
Sobolev Space ◽  
Hardy Inequality ◽  
Sufficient Conditions ◽  
Fractional Sobolev Spaces ◽  
Density Property ◽  
Smooth Functions ◽  
Fractional Sobolev Space ◽  
Compactly Supported ◽  
Bounded Set ◽  
Compactly Supported Functions

AbstractWe describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $$W^{s,p}(\Omega )$$ W s , p ( Ω ) for an open, bounded set $$\Omega \subset \mathbb {R}^{d}$$ Ω ⊂ R d . The density property is closely related to the lower and upper Assouad codimension of the boundary of $$\Omega$$ Ω . We also describe explicitly the closure of $$C_{c}^{\infty }(\Omega )$$ C c ∞ ( Ω ) in $$W^{s,p}(\Omega )$$ W s , p ( Ω ) under some mild assumptions about the geometry of $$\Omega$$ Ω . Finally, we prove a variant of a fractional order Hardy inequality.

scholarly journals Density and trace results in generalized fractal networks

2018 ◽  
Vol 52 (3) ◽  
pp. 1023-1049
Author(s):  
Serge Nicaise ◽  
Adrien Semin
Keyword(s):  
Sobolev Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Trace Operator ◽  
Compactly Supported ◽  
Weighted Trees ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

The first aim of this paper is to give different necessary and sufficient conditions that guarantee the density of the set of compactly supported functions into the Sobolev space of order one in infinite p-adic weighted trees. The second goal is to define properly a trace operator in this Sobolev space if it makes sense.

scholarly journals On the weakly-* dense subsets in $L^{\infty}(\Omega)$

2021 ◽  
Vol 16 ◽  
pp. 149
Author(s):  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Weak Convergence ◽  
Density Property ◽  
Smooth Functions ◽  
Compactly Supported

In this paper we study the density property of the compactly supported smooth functions in the space $L^{\infty}(\Omega)$. We show that this set is dense with respect to the weak-* convergence in variable spaces.

scholarly journals The extension problem for fractional Sobolev spaces with a partial vanishing trace condition

2021 ◽  
Author(s):  
Sebastian Bechtel
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Geometric Construction ◽  
Extension Problem ◽  
Fractional Sobolev Spaces ◽  
Fractional Sobolev Space ◽  
Open Set ◽  
Suitable Sense ◽  
Whole Space

AbstractWe construct whole-space extensions of functions in a fractional Sobolev space of order $$s\in (0,1)$$ s ∈ ( 0 , 1 ) and integrability $$p\in (0,\infty )$$ p ∈ ( 0 , ∞ ) on an open set O which vanish in a suitable sense on a portion D of the boundary $${{\,\mathrm{\partial \!}\,}}O$$ ∂ O of O. The set O is supposed to satisfy the so-called interior thickness condition in$${{\,\mathrm{\partial \!}\,}}O {\setminus } D$$ ∂ O \ D , which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case $$D=\emptyset $$ D = ∅ using a geometric construction.

THE DISTRIBUTIONAL -HESSIAN IN FRACTIONAL SOBOLEV SPACES

2019 ◽  
Vol 101 (3) ◽  
pp. 496-507
Author(s):  
QIANG TU ◽  
WENYI CHEN ◽  
XUETING QIU
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Weak Continuity ◽  
Fractional Sobolev Spaces ◽  
Fractional Sobolev Space ◽  
Weakly Continuous ◽  
Continuity Result ◽  
Sobolev Functions ◽  
Bounded Open Subset

We introduce the notion of a distributional $k$-Hessian ($k=2,\ldots ,n$) associated with fractional Sobolev functions on $\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in $\mathbb{R}^{n}$. We show that the distributional $k$-Hessian is weakly continuous on the fractional Sobolev space $W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional $k$-Hessian is well defined in $W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.

Functions of bounded higher variation in the fractional Sobolev spaces

2019 ◽  
Vol 22 (07) ◽  
pp. 1950056
Author(s):  
Qiang Tu ◽  
Chuanxi Wu
Keyword(s):  
Sobolev Space ◽  
Recent Work ◽  
Sobolev Spaces ◽  
Coarea Formula ◽  
Fractional Sobolev Spaces ◽  
Fractional Sobolev Space ◽  
Decomposition Property ◽  
Cartesian Currents ◽  
Definition Of ◽  
Distributional Jacobian

In this paper, we establish fine properties of functions of bounded higher variation in the framework of fractional Sobolev spaces. In particular, inspired by the recent work of Brezis–Nguyen on the distributional Jacobian, we extend the definition of functions of bounded higher variation, which defined by Jerrard–Soner in [Formula: see text], to the fractional Sobolev space [Formula: see text], and apply Cartesian currents theory to establishing general versions of coarea formula, chain rule and decomposition property.

Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes

2020 ◽  
Vol 23 (4) ◽  
pp. 967-979
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Fractional Integrals ◽  
Radon Transforms ◽  
Affine Planes ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Compactly Supported ◽  
Radial Case ◽  
Compactly Supported Functions

AbstractWe apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

An Asymptotical Stability in Variational Sense for Second Order Systems

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Dariusz Idczak ◽  
Stanisław Walczak
Keyword(s):  
Sobolev Space ◽  
Differential System ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Second Order ◽  
Functional Parameter ◽  
Asymptotical Stability ◽  
Parameter Control ◽  
Initial Condition ◽  
Second Order Systems

AbstractIn this paper, a new, variational concept of asymptotical stability of zero solution to an ordinary differential system of the second order, considered in Sobolev space, is presented. Sufficient conditions for an asymptotical stability in a variational sense with respect to initial condition and functional parameter (control) are given. Relation to the classical asymptotical stability is illustrated.

scholarly journals On linear independence for integer translates of a finite number of functions

1993 ◽  
Vol 36 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Rong-Qing Jia ◽  
Charles A. Micchelli
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Partial Difference ◽  
Integer Translates ◽  
First Case ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

scholarly journals INVERSE SPECTRAL PROBLEMS FOR STURM–LIOUVILLE OPERATORS WITH SINGULAR POTENTIALS. IV. POTENTIALS IN THE SOBOLEV SPACE SCALE

2006 ◽  
Vol 49 (2) ◽  
pp. 309-329 ◽  
Author(s):  
Rostyslav O. Hryniv ◽  
Yaroslav V. Mykytyuk
Keyword(s):  
Sobolev Space ◽  
Spectral Data ◽  
Sufficient Conditions ◽  
Inverse Spectral Problems ◽  
Singular Potentials ◽  
Spectral Problems ◽  
Sturm Liouville ◽  
Inverse Spectral ◽  
Necessary And Sufficient ◽  
Liouville Operators

AbstractWe solve the inverse spectral problems for the class of Sturm–Liouville operators with singular real-valued potentials from the Sobolev space $W^{s-1}_2(0,1)$, $s\in[0,1]$. The potential is recovered from two spectra or from one spectrum and the norming constants. Necessary and sufficient conditions for the spectral data to correspond to a potential in $W^{s-1}_2(0,1)$ are established.

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