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
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