Upper gradients and Sobolev spaces on metric spaces

The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions…


1 Introduction
1.1 The Laplace equation
1.2 The Sobolev space associated with the Laplace equation
1.3 The p-Laplace equation and its Sobolev space
1.4 The Sobolev space and how to generalize it
1.5 The aim of this thesis
1.6 Outline of this thesis
2 Preliminaries
2.1 Metric spaces and norms
2.2 Open and closed sets
2.3 The extended reals
2.4 A special constant
2.5 Lipschitz functions
2.6 Measure theory
2.7 The measure µ
2.8 Integration
2.9 Equivalence relations
2.10 Borel functions
3 Curves
3.1 Rectifiable curves
3.2 Arc length parameterization
3.3 Modulus of curve families
4 Finding a replacement for the derivative
4.1 The upper gradient
4.2 The minimal weak upper gradient
4.3 Absolute continuity
5 Newtonian spaces – Sobolev spaces on metric spaces
5.1 The Newtonian Space
5.2 Capacity
5.3 Density of Lipschitz functions
6 Final remarks

Author: Färm, David

Source: Linkoping University

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