10 PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ

We apply the quasilinear Fredholm degree to the study of fully non-

linear elliptic boundary value problems with general boundary conditions

satisfying the Shapiro-LopatinskiJ conditions. We consider the boundary

value problem

?k n

f(x,u(x) D u(x)) =0, x € n Q R

g (x,u(x),...,Dmiu(x)) = 0, x € 6Q , lik. (BVP)

Associated to (BVP) is its family of formal linearizations:

2(u)v = V fa(x,u(x),...,D2ku{x))Dav(x), x € Q

|a|2k

B i ( u ) v = X gi a(x'u(x) Dmiu(x))D°v(x),

|a|mi

(LBVP)

x € dQ, 1 i k.

2k

Our fundamental assumption amounts to the assertion that for u€C (Q),

(£(u),B (u),...,B (u)) defines an elliptic boundary value problem of order

(2k,m ,. . . ,IIL) , the m.' s are distinct, each m. 2k, and (B (u),.. . ,B (u))

satisfy the Shapiro-Lopatinskij covering conditions with respect to £(u).

2k

For u € C (Q) and x € Q, define

F(uMx) = (f(x

D2ku(x)),gl(x,...,Dmiu(x))

gk(x,. . .

,Dmku(x))).

First, in the context of Sobolev spaces, we show, using an idea of

2k+2+s-m.

Babin [Ba], that if f is C , with s n, and each gi is C \

we can formulate (BVP) directly as the zero of the mapping