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Optimal Regularity and the Free Boundaryin the Parabolic Signorini Problem

About this Title

Donatella Danielli, Nicola Garofalo, Arshak Petrosyan and Tung To

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 249, Number 1181
ISBNs: 978-1-4704-2547-0 (print); 978-1-4704-4129-6 (online)
Published electronically: August 8, 2017
Keywords:Free boundary problem, parabolic Signorini problem, evolutionary variational inequality, Almgren’s frequency formula, Caffarelli’s monotonicity formula, Weiss’s monotonicity formula, Monneau’s monotonicity formula, optimal regularity, regularity of free boundary, singular set

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Notation and preliminaries
  • Chapter 3. Known existence and regularity results
  • Chapter 4. Classes of solutions
  • Chapter 5. Estimates in Gaussian spaces
  • Chapter 6. The generalized frequency function
  • Chapter 7. Existence and homogeneity of blowups
  • Chapter 8. Homogeneous global solutions
  • Chapter 9. Optimal regularity of solutions
  • Chapter 10. Classification of free boundary points
  • Chapter 11. Free boundary: Regular set
  • Chapter 12. Free boundary: Singular set
  • Chapter 13. Weiss and Monneau type monotonicity formulas
  • Chapter 14. Structure of the singular set
  • Appendix A. Estimates in Gaussian spaces: Proofs
  • Appendix B. Parabolic Whitney’s extension theorem


We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.

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