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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

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$68
Individual Members: US$40.80
Institutional Members: US$54.40
Order Code: MEMO/182/858
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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.


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
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