Estimations polynomiales pour les problèmes de transmission sur des domaines à bords plats

Abstract

Cet article traite des opérateurs elliptiques du second ordre P dont les coefficients sont vus comme des variables aléatoires. Son but est d'obtenir des estimations de leurs solutions qui soient polynomiales dans les coefficients. Ces estimations sont utiles pour la quantification de l'incertitude. Dans cet article, nous traitons le cas dans lequel le bord et l'interface sont parallèles aux hyperplans {x m = 0} ⊂ R m .

Polynomial estimates for solutions of parametric elliptic equations on complete manifolds

“…For operators in divergence form, we obtain a slightly better result in that we may allow lower regularity for a, as in [18]. A consequence of our results is the integrability of P −1 k f H k+µ for operators of divergence form of order 2m with respect to certain measures of Gaussian type on the set of coefficients a, see Theorem 5.3.…”

“…one in W ∞,∞ ∇ (M ; T M )) integrates to a global one-parameter groups of diffeomorphisms of M and of automorphisms of our Sobolev spaces. The forth section is devoted to the proof of the main result (Theorem 1.3) following the method from [18]. The integrability of P −1 k f H s 0 +k+µ with respect to suitable Gaussian measures on the space of coefficients is proved in the last section.…”

“…The main result is stated in the Introduction states, where we also recall some needed concepts. Section 2 contains some preliminary material, including a version of Nirenberg's trick following [6,18]. The third section is devoted to proving that a totally bounded vector field (i.e.…”

“…In this section we recall a version of Nirenberg's trick, as it is formalized in [6,18]. We write t 0 if t → 0 and t > 0.…”

“…The following lemma [6,18] will play an essential role in what follows. The version here is a simplified one compared to the ones in the aforementioned articles.…”