A proof of the trace theorem of Sobolev spaces on Lipschitz domains

Abstract

Abstract. A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on H s (∂Ω). It is proved that the trace operator is a linear bounded operator from H s (Ω) to .

Extension and trace for nonlocal operators

“…The space X D is new (X stands for eXterior). It is an analogue of the classical trace space H 1/2 (∂D) [22].…”

Vascular physiology drives functional brain networks

“…The spatial similarity of the "more vascular" networks and neuronal networks may derive from patterns in neurovascular anatomy 29 : because neuronal and vascular growth processes track each other during development 30 , remote brain regions that establish neuronal links may also establish similar vascular, astrocytic, or other glial anatomy that influences local hemodynamic regulation. In the fully developed brain, environmental factors and repetitive activities (e.g., exercise) that impact the expression of neurotrophic factors may also simultaneously alter local angiogenesis [31][32][33] , allowing for ongoing and coregulated plasticity of neuronal and vascular networks. By coordinating blood flow across brain regions that typically exhibit synchronous neuronal activity, such vascular networks would also provide the most efficient hemodynamic support for increased network metabolism.…”

A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function

“…From (6), the definition of Θ x0,x1 (7) and the definition of Ω η it follows immediately that Θ x0,x1 (0) = (x 0 , η(x 0 )), Θ x1,x2 (2) = (x 1 , η(x 1 )) and Θ x0,x1 (r) ∈ Ω η , r ∈ [0, 2]. Therefore it only remains to prove (5). We calculate…”

“…The Trace Theorem for Sobolev spaces is well-known and widely used in analysis of boundary and initial-boundary value problems in partial differential equations. Usually, for the Trace Theorem to hold, the minimal assumption is that the domain has a Lipshitz boundary (see e. g. [1,5,7]). However, when studying weak solutions to a moving boundary fluid-structure interaction (FSI) problem, domains are not necessary Lipshitz (see [2,6,9,4,13]).…”

“…The Sobolev space H s (ω), 0 < s < 1 is defined by the real interpolation method (see [1,11]). However, H s (ω) can be equipped with an equivalent, intrinsic norm (see for example [1,7]) which is also used in [5]…”