Sobolev gradients: introduction, applications, problems

Abstract

Sobolev gradient is defined and a simple example is given. A number of applications are described. A number of problems are stated, some of which are open problems for research. 1. Introduction to Sobolev Gradients Sobolev gradients are an efficient method of calculating solutions to a wide variety of systems of partial differential equations. Successful applications have been made to problems in transonic flow, Ginsburg-Landau equations for su- perconductivity, elasticity, minimal surfaces and oil-water separation problems. A basic reference is (12). This reference contains motivation and background for Sobolev gradients as well as applications up to the time of publication in 1997. The present paper will summerize more recent applications and will present a number of problems. Some of the problems are exercises, some are very open ended and some may be impossible. Suppose X is a Banach space with the property that if f is a continuous linear function from X to R, then, given c> 0, there is au nique elementh ∈ X so that