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

(L u)(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 |^{2 k} \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 and 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)

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