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Concept# Gårding's inequality

Summary

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.
Statement of the inequality
Let \Omega be a bounded, open domain in n-dimensional Euclidean space and let H^k(\Omega) denote the Sobolev space of k-times weakly differentiable functions u\colon\Omega\rightarrow\mathbb{R} with weak derivatives in L^2(\Omega). Assume that \Omega satisfies the k-extension property, i.e., that there exists a bounded linear operator E\colon H^k(\Omega)\rightarrow H^k(\mathbb{R}^n) such that Eu\vert_\Omega=u for all u\in H^k(\Omega).
Let L be a linear partial differential operator of even order 2k, written in divergence form
:(L u)(x) = \sum_{0 \leq | \alpha |, | \beta | \leq k} (-1)^{| \alpha |} \

Official source

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