We give a new proof of the existence and uniqueness of the solution of
the two-dimensional membrane equations of a linearly elastic shell wh
ose middle surface is uniformly elliptic. The existence proof relies i
n particular on a lemma that J.-L. Lions has established for proving t
he three-dimensional Kern inequality. The uniqueness proof uses the un
iqueness of the Cauchy problem due to A. Calderon, then the uniqueness
of the extension of the solution of some elliptic equations due to N.
Aronszajn.