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

The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a H\"older continuous function. More precisely, let $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ and let $\Omega_{\eta}$ be a domain which is locally subgraph of a function $\eta$. We prove that mapping $\gamma_{\eta}:u\mapsto u({\bf x},\eta({\bf x}))$ can be extended by continuity to a linear, continuous mapping from $H^1(\Omega_{\eta})$ to $H^s(\omega)$, $s<\alpha/2$. This study is motivated by analysis of fluid-structure interaction problems.

1. Introduction. 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]). FSI problems have many important applications (for example in biomechanics and aero-elasticity) and therefore have been extensively studied from the analytical, as well as numerical point of view, since the late 1990s (see e.g. [2,3,6,8,9,10,12] and the references within). In FSI problems the fluid domain is unknown, given by an elastic deformation η, and therefore one cannot assume a priori any smoothness of the domain. In [2,6,9] an energy inequality implies η ∈ H 2 (ω), ω ⊂ R 2 . From the Sobolev embeddings one can see that in this case η ∈ C 0,α (ω), α < 1, but η is not necessarily Lipschitz. Nevertheless, in Section 1.3 in [2], and Section 1.3. in [6], a version of the Trace Theorem for such domains was proved, which enables the analysis of the considered FSI problems (see also [9], Section 2).
The proof of a version of the Trace Theorem in [6] (Lemma 2) relies on Sobolev embeddings theorems and the fact that η ∈ H 2 (ω) and ω ⊂ R 2 . Even though the techniques from [6] can be generalized to a broader class of Sobolev class boundaries, the result and techniques from [6] cannot be applied to some other cases of interest in FSI problems, for example to the coupling of 2D fluid flow with the 1D wave equation, where we only have η ∈ H 1 (ω) (see [4,13]) The purpose of this note is to fill that gap and generalize that result for ω ⊂ R n−1 , n > 1, and arbitrary Hölder continuous functions η. Hence, we prove a version of the Trace Theorem for a domain which is locally a subgraph of a Hölder continuous function. We use real interpolation theory (see [11]) and intrinsic norms for H s spaces, where s in not an integer.
2. Notation and Preliminaries. Let n ∈ N, n ≥ 2. Let ω ⊂ R n−1 be a Lipschitz domain and let 0 < α < 1. Furthermore, let η satisfy the following conditions: We consider the following domain with its upper boundary We define the trace operator γ η : In [2] (Lemma 1) it has been proven that γ η can be extended by continuity to an operator γ η : . This result holds with an assumption that η is only continuous. Our goal is to extend this result in a way to show that Im(γ η ) is a subspace of H s (ω), for some s > 0, when η is a Hölder continuous function.

Remark 1.
Notice that γ η is not a classical trace operator because γ η (u) is a function defined on ω, whereas the classical trace would be defined on the upper part of the boundary, Γ η . However, this version of a trace operator is exactly what one needs in analysis of FSI problems. Namely, in the FSI setting the Trace Theorem is applied to fluid velocity which, at the interface, equals the structure velocity, where the structure velocity is defined on a Lagrangian domain (in our notation ω).
3. Statement and Proof of the result.
Theorem 3.1. Let α < 1 and let η be such that conditions (1) are satisfied. Then operator γ η , defined by (2), can be extended by continuity to a linear operator from We split the main part of the proof into two Lemmas. The main idea of the proof is to transform a function defined on Ω η to a function defined on ω × (0, 1) and to apply classical Trace Theorem to a function defined on the domain ω × (0, 1). Throughout this proof C will denote a generic positive constant that depends only on ω, η and α.
Let us define function space (see [11], p. 10): where 0 < s < 1. Our goal is to proveū ∈ W (0, 1; s). However, before that we need to prove the following technical Lemma: For every x 0 , x 1 ∈ ω, there exists a piece-wise smooth curve parameterized by where C does not depend on x 0 , x 1 .
Proof. First we define x r as a convex combination of x 0 and x 1 : Furthermore we define y r in the following way: By using Hölder continuity of η we get Therefore curve (x r , y r ) stays bellow the graph of η for r ∈ [0, 1]. Now, let us consider whether this curve intersects the hyper-plane x n = η min . Since y r is a strictly decreasing function in r, we distinguish between the two separate cases.
Case 2: There exists r 0 ∈ (0, 1) such that y r = η min . In this case we define Θ x0,x1 in the following way: Analogous calculation as in Case 1 shows that estimate (5) is valid in this case as well. This completes the proof of the Lemma. Now we are ready to prove the following lemma: Let u ∈ H 1 (Ω η ) and let 0 < s < α. Thenū ∈ W (0, 1; s), whereū is defined by formula (4).