Friedrichs's inequality explained

In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the L^p norm of a function using L^p 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. Friedrichs's inequality generalizes the Poincaré–Wirtinger inequality, which deals with the case k = 1.

Statement of the inequality

Let

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

= k

\| \mathrm^ u \|_^p \right)^.

In the above

\| ⋅ \| _{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 $$\backslash mathrm^\backslash alpha\; u\; =\; \backslash frac.$$

See also

• Poincaré inequality

References

• Book: Rektorys, Karel . The Friedrichs Inequality. The Poincaré inequality . Variational Methods in Mathematics, Science and Engineering . Dordrecht . Reidel . 1977 . 2nd . 2001 . 1-4020-0297-1 . 188–198 . https://books.google.com/books?id=Kob1CAAAQBAJ&pg=PA188 .