The aim of the paper is to give a method to solve boundary value problems associated to the Helmholtz equation and to the operator of elasticity. We transform these problems in problems on the boundary Gamma of an open set of R3. After introducing a symplectic form on H1,2(G) x H-1,2(G) we obtain the adjoint of the boundary operator employed. Then the boundary problem has a solution if and only if the boundary conditions are orthogonal, for this bilinear form, to the elements of the kernel, in a good space, of the adjoint operator. We illustrate this result for a mixed problem for the Helmholtz equation (th. II.3) and the Dirichlet problem for elasticity (th. III.2), but there exist natural generalizations.
@article{urn:eudml:doc:44314,
title = {Une m\'ethode int\'egrale de fronti\`ere. Application au Laplacien et \`a l'\'elasticit\'e.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {4},
year = {1991},
pages = {265-278},
zbl = {0768.35017},
mrnumber = {MR1145699},
language = {fr},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44314}
}
Lacroix, Marie-Thérèse. Une méthode intégrale de frontière. Application au Laplacien et à l'élasticité.. Revista Matemática de la Universidad Complutense de Madrid, Tome 4 (1991) pp. 265-278. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44314/