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, Th eorie du potentiel scalaire simple couche est d erivable et il vient l'expression 8x 2 R 3 n S
, z est un point r egulier. Dans ce cas, il existe un voisinage de S en z r egulier et l'int egrale de noyau DG(y x) est trait ee de la mani ere suivante
, div S n(y) n(y) dS y + Z S\B(z;") div S n(y)G(y x) n(y)
, 1 Notons que nous utilisons la formule d'int egration par parties (II.19) qui n'est valable que pour des surfaces de Liapunov. Dans la paragraphe II.1, nous avons montrer que l'extension de (II.19)
, est physiquement pas envisageable et il doit n ec essairement exister des techniques pour contourner ce probl eme. En eeet il existe des solutions analytiques a des probl emes pr esentant des singularit es g eom etriques (le cas du cylindre en torsion est trait e dans le chapitre VI et dont le gradient ne s'annule pas aux points irr eguliers
, = () + r ^ (r ^ ()) et en remarquant que K 1 (n) = 0, il vient nalement que rK 2 (x)
, On proc ede ensuite a une int egration par parties en appliquant la formule de Stokes (II.20) et il vient nalement l'expression du gradient du potentiel de double couche 8x 2 R 3 n S
, LL a encore, nous utilisons l'hypoth ese de r egularit e de la surface au point z consid er e
, 4 Prolongement par continuit e en des points singuliers L'obtention d'une repr esentation int egrale aux points singuliers n'est pas imm ediate. Tout d'abord, il n'est pas possible de d eenir la classe de fonctions Di 1 (S) pour S 2 C 0;1 par la d eenition (II.3) Comme mentionn e dans la remarque (A.1), l'approche consistant a isoler la singularit e au point z au travers d'une int egrale de contour ne contenant plus z
, Prolongement par continuit e des potentiels ne plus consid erer les deux potentiels de mani ere s epar ee. Rappelons que toute solution de l' equation de poisson u(x) = 0 pour x 2 R 3 n S admet la repr esentation int egrale suivante 8x 2 R 3 n S
, on d ecompose de noyau rG(y x) selon sa composante normale et surfacique. Il vient alors en posant H(y; y x) = rG(t x):n(y), p.1
, y x) (y) dS y Z S H(y; y x) (y)n(y)
, y) de classe Di au point singulier. En eeet la seconde int egrale n'est autre que le potentiel de double couche de h(y) dont on conna^ t le comportement au voisinage de S. Le premier terme a la propri et e tr es int eressante de faire intervenir le noyau RG(yx) Ainsi soit z 2 S et supposons que h(y, Les techniques classiques de r egularisation
, Th eorie du potentiel scalaire