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 ''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

:$(L u)(x) = \sum_ (-1)^ \mathrm^ \left( A_ (x) \mathrm^ u(x) \right),$

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

:$\sum_ \xi^ A_ (x) \xi^ > \theta , \xi , ^ \mbox x \in \Omega, \xi \in \mathbb^ \setminus \.$

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

:$A_ \in L^ (\Omega) \mbox , \alpha , , , \beta , \leq k.$

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

:$B$, u+ G \, u \, _^ \geq C \, u \, _^ \mbox u \in H_^ (\Omega),

where

:$B$, u= \sum_ \int_ A_ (x) \mathrm^ u(x) \mathrm^ v(x) \, \mathrm 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'' ∈ ''L''²(Ω) the Poisson equation

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

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''₀¹(Ω) such that

:$B$, v= \langle f, v \rangle \mbox v \in H_^ (\Omega),

where

:$B$, v= \int_ \nabla u(x) \cdot \nabla v(x) \, \mathrm x,
:$\langle f, v \rangle = \int_ f(x) v(x) \, \mathrm x.$

The Lax–Milgram lemma ensures that if the bilinear form ''B'' is both continuous and elliptic with respect to the norm on ''H''₀¹(Ω), then, for each ''f'' ∈ ''L''²(Ω), a unique solution ''u'' must exist in ''H''₀¹(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\geq C \, u \, _^ - G \, u \, _^ \mbox u \in H_^ (\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\geq K \, u \, _^ \mbox u \in H_^ (\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 ''L''² norm of the gradient.

References

* (Theorem 9.17)

{{DEFAULTSORT:Garding's inequality
Theorems in functional analysis
Inequalities
Partial differential equations
Sobolev spaces