Sobolev embedding theorem for irregular domains and discontinuity of \(p \to p^*(p,n)\) (Q286638)

Summary: For a domain \(\Omega \subset \mathbb R^n\) we denote \[ q_\Omega(p):= \sup \{r\in[1,\infty]; \mathrm { for all } f:\Omega \to \mathbb{R}:(f \in W^{1,p}(\Omega) \Rightarrow f\in L^r (\Omega)) \}. \] Let \(p_0 \in [2,\infty)\). We construct a domain \(\Omega \subset \mathbb R^2\) such that \(q_ \Omega(p)\) is discontinuous at \(p_0\).