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 n -dimensional Euclidean space and let H ^{k(\Omega)} denote the Sobolev space of k -times weakly differentiable functions u\colon\Omega → R with weak derivatives in L ^{2(\Omega)} . Assume that \Omega satisfies the k -extension property, i.e., that there exists a bounded linear operator E\colon H ^{k(\Omega) →} H ^{k(R} ^{n)} such that Eu\vert _{\Omega=u} for all u\in H ^{k(\Omega)} .

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

(L u)(x) = \sum _{0} ( -1 ) ^{|} D ^{\alpha} \left( A _{\alpha} (x) D ^{\beta} u(x) \right),

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

\sum _{|} \xi ^{\alpha} A _{\alpha} (x) \xi ^{\beta} > \theta | \xi | ^{2} forall x \in \Omega, \xi \in 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} \in L ^{infty} (\Omega) forall | \alpha |, | \beta | \leq k.

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

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

where

B[v, u] = \sum _{0} \int _{\Omega} A _{\alpha} (x) D ^{\alpha} u(x) D ^{\beta} 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

\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 forall v \in H ^{1} _{0} (\Omega),

where

B[u, v] = \int _{\Omega} \nabla u(x) ⋅ \nabla v(x) d x, \langle f, v \rangle = \int _{\Omega} 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] \geq C \| u \| ^{2} _{H ^{1} (\Omega)} - G \| u \| ^{2} _{L ^{2} (\Omega)} forall u \in H ^{1} _{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] \geq K \| u \| ^{2} _{H ^{1} (\Omega)} forall u \in H ^{1} _{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)