Sobolev spaces on multiple cones

Auscher, Pascal; Badr, Nadine

doi:10.5802/afst.1264

Cited by 1 publication

(3 citation statements)

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“…As mentioned in [7], Theorem 10 can be immediately extended in R n to the case of a union of two halfcones sharing the same vertex, separated by a hyperplane passing through the vertex and not containing any direction of the boundaries.…”

Section: A Particular Geometry With R = R ⋆ θmentioning

confidence: 94%

“…Theorems 10 and 11 below will play an important role in the proof of Theorem 9. We start by recalling an extension result for multiple cones from [7].…”

Section: A Particular Geometry With R = R ⋆ θmentioning

confidence: 99%

“…5 -A particular geometry with r = r ⋆ θ  Remark 16. The construction in [7] is such that if v is radial (resp. constant) in C ∩ B(0, R), then Λv is radial (resp.…”

Section: A Particular Geometry With R = R ⋆ θmentioning

confidence: 99%

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A Transmission Problem Across a Fractal Self-Similar Interface

Achdou¹,

Deheuvels²

2016

Multiscale Model. Simul.

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We consider a transmission problem in which the interior domain has infinitely ramified structures. Transmission between the interior and exterior domains occurs only at the fractal component of the interface between the interior and exterior domains. We also consider the sequence of the transmission problems in which the interior domain is obtained by stopping the self-similar construction after a finite number of steps; the transmission condition is then posed on a prefractal approximation of the fractal interface. We prove the convergence in the sense of Mosco of the energy forms associated with these problems to the energy form of the limit problem. In particular, this implies the convergence of the solutions of the approximated problems to the solution of the problem with fractal interface. The proof relies in particular on an extension property. Emphasis is put on the geometry of the ramified domain. The convergence result is obtained when the fractal interface has no self-contact, and in a particular geometry with self-contacts, for which an extension result is proved.

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Section: A Particular Geometry With R = R ⋆ θmentioning

confidence: 94%

“…Theorems 10 and 11 below will play an important role in the proof of Theorem 9. We start by recalling an extension result for multiple cones from [7].…”

Section: A Particular Geometry With R = R ⋆ θmentioning

confidence: 99%