Gårding's inequality explained

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

\Omega

be a bounded, open domain in

-dimensional Euclidean space and let

H^k(\Omega)

denote the Sobolev space of

-times weakly differentiable functions

u\colon\Omega → R

with weak derivatives in

L^2(\Omega)

. Assume that

\Omega

satisfies the

-extension property, i.e., that there exists a bounded linear operator

E\colonH^{k(\Omega) →}H^k(Rⁿ⁾

such that

Eu\vert_\Omega=u

for all

u\inH^k(\Omega)

Let L be a linear partial differential operator of even order 2k, written in divergence form

(Lu)(x)=\sum₀(-1)^|D^\alpha\left(A_\alpha(x)D^\betau(x)\right),

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

\sum_|\xi^\alphaA_\alpha(x)\xi^\beta>\theta|\xi|²forallx\in\Omega,\xi\inRⁿ\setminus\{0\}.

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

A_\alpha\inL^infty(\Omega)forall|\alpha|,|\beta|\leqk.

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

B[u,u]+G\|u

	2
\\|
	L²(\Omega)

\geqC\|u

	2
\\|
	H^k(\Omega)

forallu\in

	k
H
	0

(\Omega),

where

B[v,u]=\sum₀\int_\OmegaA_\alpha(x)D^\alphau(x)D^\betav(x)dx

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

\begin{cases}-\Deltau(x)=f(x),&x\in\Omega;\ u(x)=0,&x\in\partial\Omega;\end{cases}

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]=\langlef,v\rangleforallv\in

	1
H
	0

(\Omega),

where

B[u,v]=\int_\Omega\nablau(x) ⋅ \nablav(x)dx,

\langlef,v\rangle=\int_\Omegaf(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]\geqC\|u

	2
\\|
	H¹(\Omega)

-G\|u

	2
\\|
	L²(\Omega)

forallu\in

	1
H
	0

(\Omega).

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]\geqK\|u

	2
\\|
	H¹(\Omega)

forallu\in

	1
H
	0

(\Omega),

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

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