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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.

Contents

Statement of the inequality

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

( L u ) ( x ) = ∑ 0 ≤ | α | , | β | ≤ k ( − 1 ) | α | D α ( A α β ( x ) D β u ( x ) ) ,

and suppose that L is uniformly elliptic, i.e., there exists a constant θ > 0 such that

∑ | α | , | β | = k ξ α A α β ( x ) ξ β > θ | ξ | 2 k  for all  x ∈ Ω , ξ ∈ R n ∖ { 0 } .

Finally, suppose that the coefficients Aαβ are bounded, continuous functions on the closure of Ω for |α| = |β| = k and that

A α β ∈ L ∞ ( Ω )  for all  | α | , | β | ≤ k .

Then Gårding's inequality holds: there exist constants C > 0 and G ≥ 0

B [ u , u ] + G ∥ u ∥ L 2 ( Ω ) 2 ≥ C ∥ u ∥ H k ( Ω ) 2  for all  u ∈ H 0 k ( Ω ) ,

where

B [ v , u ] = ∑ 0 ≤ | α | , | β | ≤ k ∫ Ω A α β ( x ) D α u ( x ) D β v ( x ) d x

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. (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

{ − Δ u ( x ) = f ( x ) , x ∈ Ω ; u ( x ) = 0 , x ∈ ∂ Ω ;

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

B [ u , v ] = ⟨ f , v ⟩  for all  v ∈ H 0 1 ( Ω ) ,

where

B [ u , v ] = ∫ Ω ∇ u ( x ) ⋅ ∇ v ( x ) d x , ⟨ f , v ⟩ = ∫ Ω f ( x ) v ( x ) d x .

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

B [ u , u ] ≥ C ∥ u ∥ H 1 ( Ω ) 2 − G ∥ u ∥ L 2 ( Ω ) 2  for all  u ∈ H 0 1 ( Ω ) .

Applying the Poincaré inequality allows the two terms on the right-hand side to be combined, yielding a new constant K > 0 with

B [ u , u ] ≥ K ∥ u ∥ H 1 ( Ω ) 2  for all  u ∈ H 0 1 ( Ω ) ,

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

Gårding's inequality Wikipedia
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