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Flat Level Set Regularity of $$p$$-Laplace Phase Transitions
Enrico Valdinoci, Università di Roma Tor Vertaga, Rome, Italy, Berardino Sciunzi, Università di Roma Tor Vergata, Rome, Italy, and Vasile Ovidiu Savin, University of California, Berkeley, CA
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Memoirs of the American Mathematical Society
2006; 144 pp; softcover
Volume: 182
ISBN-10: 0-8218-3910-1
ISBN-13: 978-0-8218-3910-2
List Price: US$64 Individual Members: US$38.40
Institutional Members: US\$51.20
Order Code: MEMO/182/858

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.

• 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