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
such that
Let L be a linear partial differential operator of even order k, written in divergence form
and suppose that L is uniformly elliptic, i.e., there exists a constant θ > 0 such that
Finally, suppose that the coefficients Aαβ are bounded, continuous functions on the closure of Ω for |α| = |β| = k and that
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),](../../../math/e/7/b/e7b59b9fa7e224e5c5a2f3a9acbb2529.png)
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](../../../math/8/5/8/85865f3d4c023b57375e48362a3f1fda.png)
is the bilinear form associated to the operator L.
[edit] Reference
- Rennardy, M., and Rogers, R.C. (1993). An Introduction to Partial Differential Equations. New York, NY: Springer-Verlag. ISBN 0-387-9752-2.
