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 $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.

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.

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.

Distance Functions and Orlicz-Sobolev Spaces

1986 ◽  
Vol 38 (5) ◽  
pp. 1181-1198 ◽  
Author(s):  
D. E. Edmunds ◽  
R. M. Edmunds
Keyword(s):  
Sobolev Space ◽  
Distance Function ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Distance Functions ◽  
The Mean ◽  
Image Position

Let ∧ be a bounded, non-empty, open subset of Rn and given any x in Rn, letlet k ∊ N and suppose that p ∞ (1, ∞). It is known (c.f. e.g. [4]) that if u belongs to the Sobolev space WKp(∧) and u/dk ∊ Lp(∧), then . Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out thatUnder appropriate smoothness conditions on the boundary of ∧, m and d are equivalent, and thus may in this case be characterized as the subspace of W1,2(∧) consisting of all functions u ∊ W1,2(∧) such that u/d ∊ L2(∧). Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that

scholarly journals Effective energy integral functionals for thin films in the Orlicz–Sobolev space setting

2013 ◽  
Vol 46 (3) ◽  
Author(s):  
Włodzimierz Laskowski ◽  
Hong Thai Nguyen
Keyword(s):  
Thin Film ◽  
Thin Films ◽  
Sobolev Space ◽  
Open Subset ◽  
Integral Functionals ◽  
Effective Energy ◽  
Energy Integral ◽  
Space Setting ◽  
Bounded Open Subset

AbstractWe consider an elastic thin film as a bounded open subset

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.

EXISTENCE OF FINITE ENERGY SOLUTIONS FOR ELLIPTIC SYSTEMS WITH L1-VALUED NONLINEARITIES

2008 ◽  
Vol 18 (05) ◽  
pp. 669-687 ◽  
Author(s):  
LUCIO BOCCARDO ◽  
LUIGI ORSINA ◽  
ALESSIO PORRETTA
Keyword(s):  
Elliptic System ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Finite Energy ◽  
Carathéodory Functions ◽  
Bounded Open Subset ◽  
Existence And Regularity Results ◽  
Regularity Results

In this paper, we are going to study the following elliptic system: [Formula: see text] where Ω is a bounded open subset of ℝN, a(x, s) and b(x, s) are positive and coercive Carathéodory functions, and f ∈ LM(Ω). The main purpose of this paper is to prove existence and regularity results with an improved regularity of the function z in the class of Sobolev spaces, and the existence of solutions (u, z) both with finite energy.

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.

Some uniqueness results for singular elliptic problems

2015 ◽  
Vol 26 (03) ◽  
pp. 1550026 ◽  
Author(s):  
L. Caso ◽  
R. D'Ambrosio
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Weighted Sobolev Spaces ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
Dirichlet Problems ◽  
Uniqueness Results ◽  
Singular Data

We prove some uniqueness results for Dirichlet problems for second-order linear elliptic partial differential equations in non-divergence form with singular data in suitable weighted Sobolev spaces, on an open subset Ω of ℝn, n ≥ 2, not necessarily bounded or regular.

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.

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