Assume \(1\le p < n\) and let \(p^*\) be the Sobolev conjugate of \(p\). Let \(U\subset \mathbb{R}^n\) be a bounded open set. Then there exists a constant \(C\) depending on \(U\), \(n\) and \(p\) such that

\begin{equation*} \lVert u\rVert_{L^{p^*}(U)}\le C \lVert Du\rVert_{L^p(U)} \end{equation*}

for all \(u\in W^{1,p}_0(U)\). [1, 5.6 Theorem 3]

Proof idea Link to heading

According to the definition of \(W^{1,p}_0(U)\) approximate \(u\) with test functions. Obtain the result using (0x6745c708) .

References Link to heading

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