Friedrichs' inequality

In mathematics, Friedrichs' inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent.

Statement of the inequality

Let Ω be a bounded subset of Euclidean space Rn with diameter d. Suppose that u : Ω → R lies in the Sobolev space $W_{0}^{k, p} (\Omega)$ (i.e. u lies in Wk,p(Ω) and the trace of u is zero). Then

$\| u \|_{L^{p} (\Omega)} \leq d^{k} \left( \sum_{| \alpha | = k} \| \mathrm{D}^{\alpha} u \|_{L^{p} (\Omega)}^{p} \right)^{1/p}.$

In the above

• $\| \cdot \|_{L^{p} (\Omega)}$ denotes the Lp norm;
• α = (α1, ..., αn) is a multi-index with norm |α| = α1 + ... + αn;
• Dαu is the mixed partial derivative
$\mathrm{D}^{\alpha} u = \frac{\partial^{| \alpha |} u}{\partial_{x_{1}}^{\alpha_{1}} \cdots \partial_{x_{n}}^{\alpha_{n}} }.$