Failure of first-order Sobolev embedding into Linfinity in two dimensions
= Failure of first-order Sobolev embedding into Linfinity in two dimensions
{title2=$H^1(U)\not\hookrightarrow L^\infty(U)$}
On a two-dimensional domain containing the origin, a logarithmic singularity such as $u(r)=(\log(1/r))^\alpha$ with $0<\alpha<1/2$ belongs locally to $H^1$ but is unbounded. Thus the critical first-order <Sobolev space> does not embed into $L^\infty$.
= Sobolev embedding
{synonym}