Friedrichs' inequality

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In mathematics, Friedrichs' 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.

[edit] 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 W^k,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

$\| - \|_{L^{p} (\Omega)}$ 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

$\mathrm{D}^{\alpha} u = \frac{\partial^{| \alpha |} u}{\partial_{x_{1}}^{\alpha_{1}} \cdots \partial_{x_{n}}^{\alpha_{n}} }.$

A very related result is the Poincaré inequality.

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Categories: Mathematical analysis stubs | Sobolev spaces | Inequalities

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