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The various equations at the surfaces and triple contact lines of adeformable body are obtained from a variational condition, by applying Green'sformula in the whole space and on the Riemannian surfaces. The surfaceequations are similar to the Cauchy's equations for the volume, but involve aspecial definition of the 'divergence' (tensorial product of the covariantderivatives on the surface and the whole space). The normal component of thedivergence equation generalizes the Laplace's equation for a fluid-fluidinterface. Assuming that Green's formula remains valid at the contact line(despite the singularity), two equations are obtained at this line. The firstone expresses that the fluid-fluid surface tension is equilibrated by the twosurface stresses (and not by the volume stresses of the body) and suggests afinite displacement at this line (contrary to the infinite-displacementsolution of classical elasticity, in which the surface properties are not takeninto account). The second equation represents a strong modification of Young'scapillary equation. The validity of Green's formula and the existence of afinite-displacement solution are justified with an explicit example offinite-displacement solution in the simple case of a half-space elastic solidbounded by a plane. The solution satisfies the contact line equations and itselastic energy is finite (whereas it is infinite for the classical elasticsolution). The strain tensor components generally have different limits whenapproaching the contact line under different directions. Although Green'sformula cannot be directly applied, because the stress tensor components do notbelong to the Sobolev space H1(V), it is shown that this formula remains valid.As a consequence, there is no contribution of the volume stresses at thecontact line. The validity of Green's formula plays a central role in thetheory.

Green's formula and singularity at a triple contact line. Example of finite-displacement solution