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Regularity of Finite Energy Deformations

Daniel Rosenblatt

Abstract

We review results of {Š}verák, and Goldstein-Hajłasz-Pakzad on how to show the continuity of functions in a critical Sobolev space with positive Jacobian. In the final chapter we expand on the theory of $VMO$ functions, showing a version of the change of variables theorem in this regularity, which generalizes the first step in {Š}verák's proof to a co-dimension 1 case.

Regularity of Finite Energy Deformations

Abstract

We review results of {Š}verák, and Goldstein-Hajłasz-Pakzad on how to show the continuity of functions in a critical Sobolev space with positive Jacobian. In the final chapter we expand on the theory of functions, showing a version of the change of variables theorem in this regularity, which generalizes the first step in {Š}verák's proof to a co-dimension 1 case.
Paper Structure (19 sections, 33 theorems, 141 equations, 1 figure)

This paper contains 19 sections, 33 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Introduction
  3. Notations
  4. Proof by Degree Theory
  5. Idea
  6. Introduction to Degree
  7. Generalization of Properties of Degree
  8. Useful Definitions Involving Degree
  9. Proof of Regularity by Degree
  10. Proof by Oscillation Argument
  11. Idea
  12. Oscillation Inequality
  13. Proof of Regularity
  14. Further Results - $VMO$ Theory
  15. Objectives
  16. ...and 4 more sections

Key Result

Theorem 2.1

milnor Given some $f \in C^1\left(\Omega\right)$, if we define $S_f=f\left(\left\{x\in \Omega \mid \det \nabla f\left(x\right)=0\right\}\right)$, we get that $\lambda\left(S_f\right)=0$.

Figures (1)

  • Figure 1: Left: the case $\rho<\sqrt{2}R$ Right: the case $\sqrt{2}R<\rho<2R$

Theorems & Definitions (125)

  • Remark
  • Theorem 2.1: Sard
  • Corollary 2.1.1
  • proof
  • Definition 2.1
  • Remark
  • Claim 2.2
  • Theorem 2.3: Properties of deg
  • Corollary 2.3.1
  • proof
  • ...and 115 more