# Integrodifference equation

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 *L*^{2}. Assume that Ω satisfies the *k*-extension property, i.e., that there exists a bounded linear operator *E* : *H*^{k}(Ω) → *H*^{k}(**R**^{n}) 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

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

Finally, suppose that the coefficients *A _{αβ}* are bounded, continuous functions on the closure of Ω for |

*α*| = |

*β*| =

*k*and that

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

where

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* ∈ *L*^{2}(Ω) the Poisson equation

where Ω is a bounded Lipschitz domain in **R**^{n}. The corresponding weak form of the problem is to find *u* in the Sobolev space *H*_{0}^{1}(Ω) such that

where

The Lax–Milgram lemma ensures that if the bilinear form *B* is both continuous and elliptic with respect to the norm on *H*_{0}^{1}(Ω), then, for each *f* ∈ *L*^{2}(Ω), a unique solution *u* must exist in *H*_{0}^{1}(Ω). The hypotheses of Gårding's inequality are easy to verify for the Laplace operator Δ, so there exist constants *C* and *G* ≥ 0

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

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 *L*^{2} norm of the gradient.

## References

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