A SPATIAL DECAY ESTIMATE FOR A NONLINEAR ELLIPTIC PROBLEM

A Dirichlet boundary value problem for a nonlinear elliptic p.d.e. is considered in a right cylinder, the boundary value on the lateral surface being independent of the axial coordinate. An inequality estimate, in terms of data, is obtained for the spatial rate of convergence as one recedes from the plane ends of the solution of the problem, to the solution of the corresponding two-dimensional solution (induced by the lateral boundary condition). The estimate is for a suitably defined, cross-sectional measure, which is positive-definite in the perturbation (i.e. the difference between the solution and that of the two-dimensional state). The estimate is obtained by establishing a differential inequality for the cross-sectional measure. The cross-sectional measure is analogous to a Liapunov functional that has been used in time-dependent, initial boundary value problems. The paper concludes with a discussion of the estimate obtained.