Generalized Solutions to Semilinear Elliptic PDE with Applications to the Lichnerowicz Equation

TL;DR

This paper develops a method to solve a generalized semilinear elliptic PDE with distributional data, using regularization, a priori bounds, and Colombeau algebra to handle irregular coefficients and establish existence of solutions.

Contribution

It introduces a novel approach combining regularization, a priori estimates, and Colombeau algebra to solve semilinear elliptic PDEs with distributional coefficients, extending classical methods.

Findings

Existence of solutions in Colombeau algebra for generalized PDEs.

Development of a priori bounds and sub-/super-solutions for regularized problems.

Application to the Lichnerowicz equation in general relativity.

Abstract

In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semilinear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori pointwise bounds and sub- and super-solutions and then apply a fixed-point argument for order-preserving maps. We then show that the net of solutions obtained through this process satisfies certain decay estimates by determining estimates for the sub- and super-solutions and by utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. We motivate this Colombeau algebra framework by first solving an ill-posed critical exponent problem. To solve this ill-posed problem, we use a collection of smooth, "approximating" problems and then use the resulting sequence of solutions and a compactness argument to obtain a solution to the original problem. This approach is modeled after the more general Colombeau framework that we develop, and it conveys the potential that solutions in these abstract spaces have for obtaining classical solutions to ill-posed nonlinear problems with irregular data.