A note on extension type theorems in heterogeneous mediums

Abstract

It is known for quite some time that the extension theorems play an important role in the homogenization of the periodic (heterogeneous) mediums. However, the construction of such extension operators depends on a reflection technique but for the functions in \(H^{l,r}(\Omega_p^\varepsilon)\) \((l>2)\) this reflection technique is not so straightforward, and would lead to a rather cumbersome anaylsis. In this work, we will give a short overview of some extension operators mapping from \(L^{r}(S;H^{l,r}(\Omega_p^\varepsilon))\cap H^{1,r}(S; H^{l,s}(\Omega_p^\varepsilon)^*) \to L^{r}(S;H^{l,r}(\Omega))\cap H^{1,r}(S; H^{l,s}(\Omega)^*)\) using a much simpler approach. This note also generalizes the previously known results to Lipschitz domains and for any \(r\in \mathbb{N}\) such that (s.t.) \(\frac{1}{r}+\frac{1}{s}=1\).