Formatted abstract

We consider the quasilinear elliptic partial differential equation
Qu(x) = a^{ij}(x, u(x), Du(x))D_{ij}u(x) + a(x, u(x), Du(x)) = 0 in Ω (0.1)
with the imposed nonlinear boundary condition
Gu(x) = g(x, u(x), Du(x)) = 0 on ∂Ω. (0.2)
A homotopy argument for the LeraySchauder degree is presented which reduces the question of existence of solutions to the above problem to that of establishing suitable a priori bounds on solutions of an approximating problem.
The above problem is rewritten as
a^{ij}(x,u, Du)D_{ij}U + a(x,u, Du) = 0 in Ω (0.3) u(x) = ω(x) on ∂Ω.
where ω(.) is a function which satisfies the requirement that
g(x,ω(x), Du(x)) = 0 on ∂Ω. (0.4)
The gradient term in the boundary condition is replaced with an approximating function of greater smoothness, allowing the application of LeraySchauder degree theory. We present simple assumptions which ensure the existence of both an upper solution and a lower solution of the differential operator Q, and impose simple conditions upon the boundary operator g(x,z,p) which allow us to use a homotopy argument to reduce the problem to two 'disconnected' halves, one of which is a simple Dirichlet problem in u with no reference to the function ω, the other an equation in ω with no reference to u or its derivatives. A reduction argument and reasonable assumptions upon g(x,z,p) then imply the existence of solutions to the approximating problems.
It is then shown that an a priori bound on the solutions to the approximating equations in a Hölder space of the form C^{1,α}(Ω) where α > 0 suffices to ensure the convergence of a subsequence of these functions to a classical solution of the original problem.
The result described here differ from the method of continuity in that the boundary ∂Ω, is assumed to be only C^{1,α}, and the monotonicity of the differential operator with respect to u is significantly relaxed. The techniques described also apply in situations where degree theory cannot be applied directly.
