Gårding's inequality

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 H^k(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 : H^k(Ω) → H^k(Rⁿ) such that (Eu)|_Ω = u for all u in H^k(Ω).

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

$(Lu)(x)=\sum _{{0\leq |\alpha |,|\beta |\leq k}}(-1)^{{|\alpha |}}{\mathrm {D}}^{{\alpha }}\left(A_{{\alpha \beta }}(x){\mathrm {D}}^{{\beta }}u(x)\right),$

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

$\sum _{{|\alpha |,|\beta |=k}}\xi ^{{\alpha }}A_{{\alpha \beta }}(x)\xi ^{{\beta }}>\theta |\xi |^{{2k}}{\mbox{ for all }}x\in \Omega ,\xi \in {\mathbb {R}}^{{n}}\setminus \{0\}.$

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

$A_{{\alpha \beta }}\in L^{{\infty }}(\Omega ){\mbox{ for all }}|\alpha |,|\beta |\leq k.$

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

$B[u,u]+G\|u\|_{{L^{{2}}(\Omega )}}^{{2}}\geq C\|u\|_{{H^{{k}}(\Omega )}}^{{2}}{\mbox{ for all }}u\in H_{{0}}^{{k}}(\Omega ),$

where

$B[v,u]=\sum _{{0\leq |\alpha |,|\beta |\leq k}}\int _{{\Omega }}A_{{\alpha \beta }}(x){\mathrm {D}}^{{\alpha }}u(x){\mathrm {D}}^{{\beta }}v(x)\,{\mathrm {d}}x$

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

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

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

$B[u,v]=\langle f,v\rangle {\mbox{ for all }}v\in H_{{0}}^{{1}}(\Omega ),$

where

$B[u,v]=\int _{{\Omega }}\nabla u(x)\cdot \nabla v(x)\,{\mathrm {d}}x,$

$\langle f,v\rangle =\int _{{\Omega }}f(x)v(x)\,{\mathrm {d}}x.$

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

$B[u,u]\geq C\|u\|_{{H^{{1}}(\Omega )}}^{{2}}-G\|u\|_{{L^{{2}}(\Omega )}}^{{2}}{\mbox{ for all }}u\in H_{{0}}^{{1}}(\Omega ).$

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

$B[u,u]\geq K\|u\|_{{H^{{1}}(\Omega )}}^{{2}}{\mbox{ for all }}u\in H_{{0}}^{{1}}(\Omega ),$

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

• 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)