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.
be a bounded, open domain in
-times weakly differentiable functions
-extension property, i.e., that there exists a bounded linear operator
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.
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 where The Lax–Milgram lemma ensures that if the bilinear form B is both continuous and elliptic with respect to the norm on H01(Ω), then, for each f ∈ L2(Ω), a unique solution u must exist in H01(Ω).
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.