Integrodifference equation

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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

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

(Lu)(x)=0|α|,|β|k(1)|α|Dα(Aαβ(x)Dβu(x)),

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

|α|,|β|=kξαAαβ(x)ξβ>θ|ξ|2k for all xΩ,ξn{0}.

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

AαβL(Ω) for all |α|,|β|k.

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

B[u,u]+GuL2(Ω)2CuHk(Ω)2 for all uH0k(Ω),

where

B[v,u]=0|α|,|β|kΩAαβ(x)Dαu(x)Dβv(x)dx

is the bilinear form associated to the operator L.

Application: the Laplace operator and the Poisson problem

As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L2(Ω) the Poisson equation

{Δu(x)=f(x),xΩ;u(x)=0,xΩ;

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 vH01(Ω),

where

B[u,v]=Ωu(x)v(x)dx,
f,v=Ωf(x)v(x)dx.

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]CuH1(Ω)2GuL2(Ω)2 for all uH01(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]KuH1(Ω)2 for all uH01(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

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