Gårding's inequality
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.
Let Ω be a bounded, open domain in n-dimensional Euclidean space and let Hk(Ω) denote the Sobolev space of k-times weakly differentiable functions u : Ω → R with weak derivatives in L2. Assume that Ω satisfies the k-extension property, i.e., that there exists a bounded linear operator E : Hk(Ω) → Hk(Rn) such that (Eu)|Ω = u for all u in Hk(Ω).
Let L be a linear partial differential operator of even order 2k, 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 > 0 and G ≥ 0
where
is the bilinear form associated to the operator L.
Be careful, in this application, Garding's Inequality seems useless here as the final result is a direct consequence of Poincaré's Inequality, or Friedrich Inequality. (See talk on the article).
As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L2(Ω) the Poisson equation
where Ω is a bounded Lipschitz domain in Rn. The corresponding weak form of the problem is to find u in the Sobolev space H01(Ω) such that
...
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