Boundary traces of functions from Sobolev spaces on a domain with a cusp
The authors study the space of traces for the functions of the Sobolev space \(W^1_p (\Omega)\), \(p \in (1,\infty)\) assuming that the domain \(\Omega \subset \mathbb R^n\), \(n \geq 3\) has inner or outer edges. Using the \textit{E. Gagliardo} results from [Rend. Sem. Mat. Univ. Padova 27, 284--305 (1957; Zbl 0087.10902)] they explicitly construct several equivalent norms to the trace norm \[ \| f \|_{TW^1_p (\Omega)} = \inf \bigl\{ \| u \|_{W^1_p (\Omega)} : u \in W^1_p (\Omega),\;u |_{\partial \Omega} = f \text{ in the sense of traces} \bigr\}. \] The authors consider in detail the special case when \(\Omega = \{x \in \mathbb R^n : x^2_1 + \ldots + x^2_{n-1} \leq \varphi (x_n)\), \(x_n \in (0,1)\}\), where \(\varphi \in C^1 ([0,1])\) is an increasing function such that \(\varphi (0) = \varphi'(0) = 0\).
scientific article; zbMATH DE number 6320