  In this paper we prove the existence and the stability of small-amplitude
quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced
Kirchoff equation with periodic boundary conditions. This is the first KAM
result for a quasi-linear wave-type equation. The main difficulties are: $(i)$
the presence of the highest order derivative in the nonlinearity which does not
allow to apply the classical KAM scheme, $(ii)$ the presence of double
resonances, due to the double multiplicity of the eigenvalues of $-
\partial_{xx}$. The proof is based on a Nash-Moser scheme in Sobolev class. The
main point concerns the invertibility of the linearized operator at any
approximate solution and the proof of tame estimates for its inverse in high
Sobolev norm. To this aim, we conjugate the linearized operator to a $2 \times
2$, time independent, block-diagonal operator. This is achieved by using {\it
changes of variables} induced by diffeomorphisms of the torus, {\it
pseudo-differential} operators and a KAM {\it reducibility} scheme in Sobolev
class.

Analysis of PDEs