Gårding inequality

An inequality of the form

$$ \| u \| _ {m} ^ {2} \leq \ c _ {1} \mathop{\rm Re} B [ u, u] = \ c _ {2} \| u \| _ {0} ^ {2} , $$

where $ u \in C _ {0} ^ \infty ( G) $ is a complex-valued function with compact support (in $ G $), $ G \subset \mathbf R ^ {n} $ is a bounded domain and

$$ B [ u, u] = \ \sum _ {| s |, | t | \leq m } \ \int\limits _ { G } a _ {st} D ^ {s} u \overline{ {D ^ {t} u }}\; dx $$

is a quadratic integral form with complex continuous coefficients $ a _ {st} $ in $ \overline{G}\; $. A sufficient condition for the Gårding inequality to be valid for any function $ u \in C _ {0} ^ \infty ( G) $ is the existence of a positive constant $ c _ {0} $ such that

$$ \mathop{\rm Re} \sum _ {| s |, | t | \leq m } a _ {st} \xi ^ {s} \xi ^ {t} \geq \ c _ {0} | \xi | ^ {2m} , $$

for any $ x \in G $ and all real vectors $ \xi = ( \xi ^ {1} \dots \xi ^ {n)} $. Formulated and proved by L. Gårding [1].

References

[1]	L. Gårding, "Dirichlet's problem for linear elliptic partial differential equations" Math. Scand. , 1 (1953) pp. 55–72
[2]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1

Comments

A sharp form of this inequality has been given by L. Hörmander. See [a1], Sects. 18.1, 18.6, and the literature quoted there.

References

[a1]	L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985)