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

Abstract

The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. More precisely, let η ∈ C 0,α (ω), 0 < α < 1 and let Ωη be a domain which is locally subgraph of a function η. We prove that mapping γη : u → u(x, η(x)) can be extended by continuity to a linear, continuous mapping from H 1 (Ωη) to H s (ω), s < α/2. This study is motivated by analysis of fluid-structure interaction problems.

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

“…It was shown in [9,28,41] that the trace operator γ Γ(t) can be extended by continuity to a linear operator from…”

“…For a precise statement of the results about "Lagrangian" trace see [41]. Now, we can define the velocity solution space defined on the moving domain in the following way:…”

“…Namely, since the fluid-structure interface Γ is not necessarily Lipschitz we need to use the results about functions defined on a domain which is a sub-graph of a Hölder continuous function, to be able to obtain certain regularity of u N and make sense of its trace on Γ, presented in [41]. First, from [41] and statement 2 in Lemma 4.1 we get that sequence (u N ) N ∈N is uniformly bounded in L 2 (0, T ; H s (Ω)), s < 1. Then, we notice that…”

“…and use the result about the trace on a domain which is not Lipschitz obtained in [41] to conclude that the sequence (v N ) N ∈N is uniformly bounded in L 2 (0, T ; H s/2 (ω)), s < 1. We complete the proof by noticing that H s (Ω) and H s/2 (ω) are compactly embedded in L 2 (Ω) and L 2 (ω), respectively, 0 < s < 1, thereby showing spatial compactness of the approximating sequences…”

Existence of a solution to a fluid–multi-layered-structure interaction problem

“…This is a more difficult case since the fourth-order flexural term ∂ 4 z , which provides higher regularity of weak solutions in the Koiter shell problem, is not present in the pure membrane model described by the linear wave equation. As a result, the analysis is more involved, and a non-standard version of the Trace Theorem (see [44] and Theorem 6.2) needs to be used to obtain the existence result.…”