The key point in the proof of (i) is to obtain λ 1 < 0 by contradiction.
The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties.
Since the key point in the proof of Theorem 3.2.1 is the center manifold function, we introduce an approximation formula of the center manifold function derived in [16].
Beside the choice of operator (25), which is naturally imposed, a crucial point in the proof of Theorem 2 is the choice of constant q in (23) to get the contractivity of the operator.
We characterize the exact observability of this system only in terms of C and of the spectral elements of the operator A. The starting point in the proof of this result is a Hautus-type test, recently obtained in Burq and Zworski (J. Amer. Soc. 17 (2004) 443 471) and Miller (J. Funct. Anal. 218 (20052005) 425 444).
The key point in the proof of this lemma is the observation that, as a consequence of (6.5), the real tangential Hessian of any defining function for a domain as in Lemma 3 is positive definite when restricted to the complex tangent space T w C ( b D ) (viewed as a vector space over the real numbers).
Finally, as in Step 3 of the same proof, one can show that there exists a suitable constant C such that | u n ′ ( t ) | < C ≤ N C ( t ) for every t ∈ [ − L, L ]. Notice that till this point in the proof of [[21], Theorem 3.1], the definition of N C, or the fact that γ > 1, were not used.
end{aligned}The key point in the proof of Theorem 5.13 is to study the minimization problem begin{aligned} min Big {P_gamma (E +frac{varepsilon }{2}|b(E)|^2+Lambda |gamma (E -Phi (s)|:,E -Phit mathbb {R}^nBig } end{aligned}and to show that for (varepsilon ) sufficiently small, and (Lambda ) large, the only minima are the half spaces (H_{nu,s}).
The crucial point in the proof of the existence of a solution to the subdifferential inclusions considered below is the following surjectivity result.
