Monotonicity with discontinuities in partial differential equations

Abstract

In this survey monotonicity theorems for linear and nonlinear partial differential equations of second order are collected, in which discontinuities in the coefficients of the differential equation or in lower and upper bounds v,w for the solution u may occur. Discontinuities at certain interfaces are considered. If v,w are continuous, but their normal derivatives not, one has interface conditions for the normal derivatives. The form of these conditions can be found from a quite general geometrical principle. Results for special types of differential equations are cited. For monotone operators of contractive type one gets often error bounds, for monotone operators of non contractive type the iteration procedure still gives a numerical method; for demonstration a nonlinear boundary value problem with several solutions is considered.