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.

Contents

Statement of the inequality

References

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 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
: mathrm{D}^{lpha} left( A_{lpha eta} (x) mathrm{D}^{eta} u(x)
ight),
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'' 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
:
is the bilinear form associated to the operator ''L''.

References

★ An introduction to partial differential equations, Renardy, Michael and Rogers, Robert C., , , Springer-Verlag, 2004, (Theorem 8.17)