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.

scholarly journals Traces of multipliers in pairs of weighted Sobolev spaces

2005 ◽  
Vol 3 (1) ◽  
pp. 91-115
Author(s):  
Vladimir Maz'ya ◽  
Tatyana Shaposhnikova
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Weighted Sobolev Space ◽  
Weighted Sobolev Spaces ◽  
Fractional Sobolev Spaces ◽  
Pointwise Multipliers

We prove that the pointwise multipliers acting in a pair of fractional Sobolev spaces form the space of boundary traces of multipliers in a pair of weighted Sobolev space of functions in a domain.

scholarly journals Sobolev spaces and hyperbolic fillings

2018 ◽  
Vol 2018 (737) ◽  
pp. 161-187 ◽  
Author(s):  
Mario Bonk ◽  
Eero Saksman
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Weak Type ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Fractional Sobolev Space ◽  
Gradient Condition

AbstractLetZbe an AhlforsQ-regular compact metric measure space, where{Q>0}. For{p>1}we introduce a new (fractional) Sobolev space{A^{p}(Z)}consisting of functions whose extensions to the hyperbolic filling ofZsatisfy a weak-type gradient condition. IfZsupports aQ-Poincaré inequality with{Q>1}, then{A^{Q}(Z)}coincides with the familiar (homogeneous) Hajłasz–Sobolev space.

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 Reproducing kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

2018 ◽  
Vol 16 (05) ◽  
pp. 693-715 ◽  
Author(s):  
Erich Novak ◽  
Mario Ullrich ◽  
Henryk Woźniakowski ◽  
Shun Zhang
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Reproducing Kernel ◽  
Absolute Error ◽  
Reproducing Kernels ◽  
Worst Case ◽  
Reproducing Kernel Formula ◽  
Positive Integers ◽  
Integration Errors ◽  
Exponentially Small

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

scholarly journals Composition in fractional Sobolev spaces

2001 ◽  
Vol 7 (2) ◽  
pp. 241-246 ◽  
Author(s):  
Haim Brezis ◽  
◽  
Petru Mironescu ◽  
Keyword(s):  
Sobolev Spaces ◽  
Fractional Sobolev Spaces

scholarly journals Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

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