Assume \(1\le p < n\) and let \(p^*\) be the Sobolev conjugate of \(p\). Suppose \(U\subset \mathbb{R}^n\) be a bounded open set and suppose \(\partial U\) is in \(C^1\). Then

\begin{equation*} W^{1,p}(U)\subset \subset L^q(U) \end{equation*}

for each \(1\le q < p^*\). [1, 5.7 Theorem 1]

Proof idea Link to heading

The prove uses the Arzelà-Ascoli theorem .

References Link to heading

1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.