Failure of first-order Sobolev embedding into Linfinity in two dimensions ($H^1(U)/\hookrightarrow L^{\infty }(U)$)

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 $L^{\infty }$.