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:
  1.  They should send polynomial on polyomials without increasing the partial degrees,
  2.  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
  1. The space  P_N  provided with the  Hm  norm
  2. 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

Pdf file (700 k)

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 !