The geometry of finite difference discretizations of semilinear elliptic operators

Abstract

Discretizations by finite differences of some semilinear elliptic equations lead to maps , given by nonlinear convex diagonal perturbations of symmetric matrices A. For natural nonlinearity classes, we consider the equation F(u) = y − tp, where t is a large positive number and p is a vector with negative coordinates. As the range of the derivative f′i of the coordinates of f encloses more eigenvalues of A, the number of solutions increases geometrically, eventually reaching 2n. This phenomenon, somewhat in contrast with behaviour associated with the Lazer–McKenna conjecture, has a very simple geometric explanation: a perturbation of a multiple fold gives rise to a function which sends connected components of its critical set to hypersurfaces with large rotation numbers with respect to vectors with very negative coordinates.Strictly speaking, the results have nothing to do with elliptic equations: they are properties of the interaction of a (self-adjoint) linear map with increasingly stronger nonlinear convex diagonal interactions.