Partial Differential Equations - Modelling and ... - ResearchGate

266 Y. Achdou
Proposition 3. If the assumptions of Proposition 2 are satisfied and if 1 2 <
α 1 2
, defining 〈·, ·〉 as the
duality pairing between V −s and V s .
Proposition 4 (Gårding inequality). If the assumptions of Proposition 2
are satisfied and if there exists a constant ψ such that ψ ≥ ψ > 0 almost
everywhere in a neighborhood of 0, then
(i) if 1 2
< α < 1, there exists a positive constant C and a non-negative
constant λ such that, for all v ∈ V α ,
〈Bv,v〉 ≥C|v| 2 V α − λ‖v‖2 L 2 (R +) ; (27)
(ii) if α ≤ 1 2 , then (27) holds for any v ∈ V s , s> 1 2
(〈·, ·〉 standing for the
duality pairing between V −s and V s ).
Consider the two situations:
1
1.
2