Solution of monotone nonlinear elliptic boundary value problems

Abstract

A bstraa. Let L be a linear uniformly elliptic second order operator. The boundary value problem is solved for the nonlinear elliptic equationis a monotone increasing function of u for each point x in the domain. A descent technique based on Newton's method is shown to yield a sequence of iterates which converges uniformly and quadratically to the solution. The convergence is independent of the choice for the initial iterate. Numerical results in two dimensions are presented.

Limiting behavior of a class of Hermitian Yang-Mills metrics, I

“…Proof. Since for each x = (x 1 , x 2 ) the function ǫ −2 e u − r 2 e −u is a monotone increasing function of u, according to [25] the boundary value problem (4.2) has a unique solution.…”

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