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.

tatement 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 "k", written in divergence form

:

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

:

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"₀¹(Ω) such that

:$B\; [u,\; v]\; =\; langle\; f,\; v\; angle\; mbox\{\; for\; all\; \}\; v\; in\; H\_\{0\}^\{1\}\; (Omega),$

where

:$B\; [u,\; v]\; =\; int\_\{Omega\}\; abla\; u(x)\; cdot\; abla\; v(x)\; ,\; mathrm\{d\}\; x,$:$langle\; f,\; v\; angle\; =\; int\_\{Omega\}\; f(x)\; v(x)\; ,\; mathrm\{d\}\; x.$

The Lax-Milgram lemma ensures that if the bilinear form "B" is both continuous and elliptic with respect to the norm on "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

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

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 "L"² norm of the gradient.

References

* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 356
isbn = 0-387-00444-0 (Theorem 8.17)