We prove a Harnack inequality for level sets of \(p\)-Laplace phase transition minimizers. In particular, if a level set is included in a flat cylinder, then, in the interior, it is included in a flatter one. The extension of a result conjectured by De Giorgi and recently proven by the third author for \(p=2\) follows.

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Table of Contents

• Introduction
• Modifications of the potential and of one-dimensional solutions
• Geometry of the touching points
• Measure theoretic results
• Estimates on the measure of the projection of the contact set
• Proof of Theorem 1.1
• Proof of Theorem 1.2
• Proof of Theorem 1.3
• Proof of Theorem 1.4
• Appendix A. Proof of the measure theoretic results
• Appendix B. Summary of elementary lemmata
• Bibliography