Assume \(n< p \le \infty\). Let \(U\subset \mathbb{R}^n\) be a bounded open set and suppose \(\partial U\) is in \(C^1\). There exists a constant \(C\) depending on \(U\), \(n\) and \(p\) such that

\begin{equation*} \lVert u\rVert_{C^{0,\gamma}(\bar{U})}\le C \lVert u\rVert_{W^{1,p}(U)} \end{equation*}

for all \(u\in W^{1,p}(U)\), where

\begin{equation*} \gamma=1-\frac{n}{p} \end{equation*}

[1, 5.6 Theorem 5].

See also Link to heading

• Morrey’s inequality on \(\mathbb{R}^n\)
• Morrey’s inequality for higher derivatives

References Link to heading

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