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Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem

About this Title

Donatella Danielli, Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, Nicola Garofalo, Dipartimento d’Ingegneria Civile e Ambientale (DICEA), Università di Padova, via Trieste 63, 35131 Padova, Italy, Arshak Petrosyan, Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 and Tung To, Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

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
MSC: Primary 35R35, 35K85

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Table of Contents


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