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 On the trace space of a Sobolev space with a radial weight

2008 ◽  
Vol 6 (3) ◽  
pp. 259-276 ◽  
Author(s):  
Helmut Abels ◽  
Miroslav Krbec ◽  
Katrin Schumacher
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Large Range ◽  
Weighted Sobolev Spaces ◽  
Full Description ◽  
Trace Space ◽  
Radial Weight ◽  
Trace Spaces

Our concern in this paper lies with trace spaces for weighted Sobolev spaces, when the weight is a power of the distance to a point at the boundary. For a large range of powers we give a full description of the trace space.

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 Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

Compact embeddings of some weighted Sobolev spaces on ℝN

1995 ◽  
Vol 117 (2) ◽  
pp. 333-338 ◽  
Author(s):  
Raffaele Chiappinelli
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Compact Embeddings ◽  
Measurable Functions ◽  
Image Position

Let ρ,ρ0,ρ1 be positive, measurable functions on ℝN. For 1 ≤ t < ∞, consider the weighted Lebesgue and Sobolev spaces

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.

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