Traces of weighted function spaces: Dyadic norms and Whitney extensions

Koskela, Pekka; Soto, Tomás; Wang, Zhuang

doi:10.1007/s11425-017-9148-6

Cited by 17 publications

(18 citation statements)

References 40 publications

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“…For Sobolev spaces with fractional order of differentiability, which can be seen as one type of nonlocal problems with infinite interaction length, the trace space and extension results are studied in [11,26,38,53]. The latter can be useful in studying nonlocal problems with non-homongeneous boundary data, such as those associated with the nonlocal Laplacian and nonlocal p-Laplacian, see for example [2-4, 11, 20, 52].…”

Section: Local Nonlocal and Fractional Modelingmentioning

confidence: 99%

“…with the classical Whitney decomposition of the half space, where the length of each cube is proportional to the distance between the cube and the boundary of the domain. This type of decomposition is also used to prove the classical and fractional extension results [26,38]. The Type I decomposition, however, has a special set W 0 which touches the boundary {0} × R d−1 and it is used later to construct extension operator for the case β < d.…”

Section: Dyadic Cubes and Whitney Type Decompositionmentioning

confidence: 99%

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Nonlocal Trace Spaces and Extension Results for Nonlocal Calculus

Du¹,

Tian²,

Wright³

et al. 2021

Preprint

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For a given Lipschitz domain Ω, it is a classical result that the trace space of W 1,p (Ω) is W 1−1/p,p (∂Ω), namely any W 1,p (Ω) function has a well-defined W 1−1/p,p (∂Ω) trace on its codimension-1 boundary ∂Ω and any W 1−1/p,p (∂Ω) function on ∂Ω can be extended to a W 1,p (Ω) function. Recently,[26] characterizes the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain R d \Ω. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical W 1−1/p,p (∂Ω) space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.

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Section: Local Nonlocal and Fractional Modelingmentioning

confidence: 99%

Section: Dyadic Cubes and Whitney Type Decompositionmentioning

confidence: 99%

Nonlocal Trace Spaces and Extension Results for Nonlocal Calculus

Du¹,

Tian²,

Wright³

et al. 2021

Preprint

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“…For other results on existence of an extension operator exists for Muckenhoupt class of weights, see e.g. [2,5,9,11] and references therein. In general, we can deal with such weight functions w which either may vanish somewhere in Ω or increase to infinity or both.…”

Section: Proposition 22 ([6]mentioning

confidence: 99%

“…On the other hand to treat the nonhomogeneous boundary conditions it requires a trace operator defined in weighted Sobolev space. The trace and extension theory in weighted Sobolev space is available (e.g., [11,9,5]) upto some restrictions to the associated weight functions. An extension result with less restrictions on the weight functions, for example, can be found by Kałamajska and the author in [2] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence of solutions of degenerate semilinear elliptic boundary value problems

Dhara¹

2021

Preprint

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“…For the case of manifolds see [12,13] and for the setting of metric spaces see [2,4,15]. Classical trace results on the Euclidean spaces can be found in [1,6,9,14,16,22,25,29,30] and studies of parabolicity on infinite networks in [26,32]. For trace results in the metric setting see [3,[18][19][20][21]33].…”

Section: Introductionmentioning

confidence: 99%