Gårding's inequality

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In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding.

[edit] Statement of the inequality

Let Ω be a bounded, open domain in n-dimensional Euclidean space. Assume that Ω satisfies the k-extension property that there exists a bounded linear map

$E : H^{k} (\Omega) \to H^{k} (\mathbb{R}^{n})$

such that

$(E u)|_{\Omega} = u \mbox{ for all } u \in H^{k} (\Omega).$

Let L be a linear partial differential operator of even order k, written in divergence form

$(L u)(x) = \sum_{0 \leq | \alpha |, | \beta | \leq k} (-1)^{| \alpha |} \mathrm{D}^{\alpha} \left( A_{\alpha \beta} (x) \mathrm{D}^{\beta} u(x) \right),$

and suppose that L is uniformly elliptic, i.e., there exists a constant θ > 0 such that

$\sum_{| \alpha |, | \beta | = k} \xi^{\alpha} A_{\alpha \beta} (x) \xi^{\beta} > \theta | \xi |^{2 k} \mbox{ for all } x \in \Omega, \xi \in \mathbb{R}^{n} \setminus \{ 0 \}.$

Finally, suppose that the coefficients A_αβ are bounded, continuous functions on the closure of Ω for |α| = |β| = k and that

$A_{\alpha \beta} \in L^{\infty} (\Omega) \mbox{ for all } | \alpha |, | \beta | \leq k.$

Then Gårding's inequality holds: there exist constants C and G ≥ 0

$B[u, u] + G \| u \|_{L^{2} (\Omega)}^{2} \geq C \| u \|_{H^{k} (\Omega)}^{2} \mbox{ for all } u \in H_{0}^{k} (\Omega),$

where

$B[v, u] = \sum_{0 \leq | \alpha |, | \beta | \leq k} (-1)^{| \alpha |} \int_{\Omega} A_{\alpha \beta} (x) \mathrm{D}^{\alpha} u(x) \mathrm{D}^{\beta} v(x) \, \mathrm{d} x$