Friedrichs's inequality

Let ${\displaystyle \Omega }$ be a bounded subset of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with diameter ${\displaystyle d}$. Suppose that ${\displaystyle u:\Omega \to \mathbb {R} }$ lies in the Sobolev space ${\displaystyle W_{0}^{k,p}(\Omega )}$, i.e., ${\displaystyle u\in W^{k,p}(\Omega )}$ and the trace of ${\displaystyle u}$ on the boundary ${\displaystyle \partial \Omega }$ is zero. Then

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

In the above

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

  

• Poincaré inequality

• Rektorys, Karel (2001) [1977]. "The Friedrichs Inequality. The Poincaré inequality". Variational Methods in Mathematics, Science and Engineering (2nd ed.). Dordrecht: Reidel. pp. 188–198. ISBN 1-4020-0297-1.