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.

Statement of the inequality

Let Ω be a bounded, open domain in n-dimensional Euclidean space and let Hk 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 → Hk such that |Ω = u for all u in Hk.
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.

Application: the Laplace operator and the Poisson problem

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..
As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L2 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 such that
where
The Lax–Milgram lemma ensures that if the bilinear form B is both continuous and elliptic with respect to the norm on H01, then, for each f ∈ L2, a unique solution u must exist in H01. The hypotheses of Gårding's inequality are easy to verify for the Laplace operator Δ, so there exist constants C and G ≥ 0
Applying the Poincaré inequality allows the two terms on the right-hand side to be combined, yielding a new constant K > 0 with
which is precisely the statement that B is elliptic. The continuity of B is even easier to see: simply apply the Cauchy–Schwarz inequality and the fact that the Sobolev norm is controlled by the L2 norm of the gradient.