Some counterexamples to the embedding theorems for Sobolev spaces (Q1422717)

The aim of the paper is to construct bounded domains \(\Omega\) in \({\mathbb R}^n\), \(n>2\), such that Sobolev spaces defined on them can give us different counterexamples to Sobolev embedding theorems. The three types of Sobolev spaces are \(V^\ell_p(\Omega)\), \(W^\ell_p(\Omega)\), \(L^\ell_p(\Omega)\), \(l=1,2,\ldots\), \(1\leq p \leq \infty\). The norm in the first spaces in the sum of \(L_p\)-norms of all gradients \(\nabla_j\), \(j=0,\ldots \ell\), whereas in the second cases we take only \(j=0\) and \(j=\ell\). In this case, one takes once more \(j=0\) and \(j=\ell\) but only those functions that are locally \(p\)-integrable, and the \(L_p\)-norm of the function is taken over some subset with compact closure contained in \(\Omega\). Examples of domains are constructed with the following properties of the Sobolev space: -- the spaces \(V^\ell_p(\Omega)\), \(W^\ell_p(\Omega)\), \(L^\ell_p(\Omega)\) do not coincided pairwise for any \(\ell=2,3,\ldots\), \(1\leq p<\infty\), -- embeddings \(V^\ell_p(\Omega)\subset L_p(\Omega)\) are not compact for any \(\ell=2,3,\ldots\), -- \(L^\ell_p(\Omega)\) is not embedded into \(L_q(\Omega)\) for any \(0<q<\infty\), -- \(L^\ell_p(\Omega)\) is embedded into \(L_\infty(\Omega)\) but it is not an algebra with pointwise multiplication, \(\ell,n>2\), \(1\leq p< \infty\), \(p\ell>n\). Some other similar examples are also considered.