Contents
Chapter 1. Introduction 1
Chapter 2. Modifications of the potential and of one-dimensional solutions 7
Chapter 3. Geometry of the touching points 31
Chapter 4. Measure theoretic results 43
Chapter 5. Estimates on the measure of the projection of the contact set 47
Chapter 6. Proof of Theorem 1.1 61
Chapter 7. Proof of Theorem 1.2 69
Chapter 8. Proof of Theorem 1.3 75
Chapter 9. Proof of Theorem 1.4 79
Appendix A. Proof of the measure theoretic results 83
A. 1. Proof of Lemma 4.1 83
A.2. Proof of Lemma 4.2 88
A.3. Proof of Lemma 4.3 127
Appendix B. Summary of elementary lemmata 135
Bibliography 143
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