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. Friedrichs's inequality generalizes the Poincaré–Wirtinger inequality, which deals with the case k = 1.

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

‖ u ‖ L p ( Ω ) ≤ d k ( ∑ | α | = k ‖ D α u ‖ L p ( Ω ) p ) 1 / p . {\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

‖ ⋅ ‖ L p ( Ω ) {\displaystyle \|\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 D α u = ∂ | α | u ∂ x 1 α 1 ⋯ ∂ x n α n . {\displaystyle \mathrm {D} ^{\alpha }u={\frac {\partial ^{|\alpha |}u}{\partial _{x_{1}}^{\alpha _{1}}\cdots \partial _{x_{n}}^{\alpha _{n}}}}.}

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