A dynamical approach to semilinear elliptic equations

2019/07/23 by Beck, Margaret, Cox, Graham, Jones, Christopher +2 · 1 citation
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.09986

Abstract

A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in ℝⁿ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.

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