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.

Statement of the inequality[edit]

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.

Application: the Laplace operator and the Poisson problem[edit]

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

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.

References[edit]

  • Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0.  (Theorem 9.17)