Polynomials in the Sobolev World
Christine Bernardi,
Monique Dauge
and
Yvon Maday
This three chapter document addresses continuity and lifting of
traces in singular domains, such as polygons and polyhedra.
Compatibility conditions at corners and edges are described. Lifting
operators are constructed on reference square and cube, with the
following two requirements:
- They should send polynomial on polyomials without increasing the partial degrees,
- They should be stable in optimal Sobolev norms, standard and weighted.
As a by-product, an important result about the interpolation
between polynomial spaces P_N with different Sobolev norms is derived:
Functional interpolation between
- The space P_N provided with the Hm norm
- The same space P_N provided with the L2 norm
yields once more the space P_N with the right Sobolev norm Hs , and with equivalence constants independent of the degree N.
Inverse inequalities are also investigated.
Preprint IRMAR 07-14, Rennes, March 2007
The prologue takes the form of a dialog between two Sobolev spaces.
And, like old cultured Russian people, they are speaking french...
When W^{s1,p1}_a1(Omega) meets his colleague
W^{s2,p2}_a2(Omega), he tells him:
«
— Je suis plus grand que toi.
— Pourquoi?
— Parce que je contiens plus de polynômes que toi.
— Idiot! Tu ne contiens aucun polynôme puisque notre Omega est borné. Et je contiens les restrictions d'exactement les mêmes polynômes que toi.
— Alors je suis plus grand que toi parce que je suis moins régulier. Mon s1 - (a1 + d)/p1 ...
— Ah! Tu as enfin compris les inégalités de notre maître Sobolev! Ce ne sont pas des
polynômes qui vont nous départager... Et il y a même encore plus fort !
— ??
— Si on interpole les espaces
de polynômes entre deux normes de Sobolev, on retrouve la bonne
norme interpolée, indépendamment du degré !
— Ils ont vraiment démontré ça, enfin ?
— Oui, j'ai tout vérifié.
— Ah ! Ca me soulage d'un grand poids !
»