Leray-Schauder degree theory and partial differential equations under nonlinear boundary conditions
Timothy R. Cranny · 1992
We consider the quasilinear elliptic partial differential equation Qu(x) = aij(x, u(x), Du(x))Diju(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 Leray-Schauder 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 re-written asaij(x,u, Du)DijU + 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 Leray-Schauder 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 Holder space of the form C1,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 C1,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.