Assume \(\frac{n}{k}< p \le \infty\). 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 \(U\), \(n\), \(k\) and \(p\) such that

\begin{equation*} \lVert u\rVert_{C^{k-\bigl\lfloor\frac{n}{p}\bigr\rfloor-1,\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=\biggl\lfloor \frac{n}{p} \biggr\rfloor + 1 - \frac{n}{p} \end{equation*}

if \(\frac{n}{p}\) is not an integer and \(0<\gamma <1\) arbitrary otherwise. [1, 5.6 Theorem 6]

See also Link to heading

• Morrey’s inequality on a bounded domain with \(C^1\)-boundary.
• Morrey’s inequality on \(\mathbb{R}^n\)

References Link to heading

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