Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem

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 th...

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Hlavní autoři: Danielli, Donatella, Garofalo, Nicola, Petrosyan, Arshak, To, Tung
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, Rhode Island American Mathematical Society 2017
Vydání:1
Edice:Memoirs of the American Mathematical Society
Témata:
Boundary value problems
Elasticity
Elasticity > Mathematical models
Mathematical physics
singular set
Almgren’s frequency formula
Caffarelli’s monotonicity formula
optimal regularity
evolutionary variational inequality
Free boundary problem
regularity of free boundary
Weiss’s monotonicity formula
parabolic Signorini problem
Monneau’s monotonicity formula
ISBN:9781470425470, 1470425475
ISSN:0065-9266, 1947-6221
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  • Introduction -- Notation and preliminaries -- Known existence and regularity results -- Classes of solutions -- Estimates in Gaussian spaces -- The generalized frequency function -- Existence and homogeneity of blowups -- Homogeneous global solutions -- Optimal regularity of solutions -- Classification of free boundary points -- Free boundary: Regular set -- Free boundary: Singular set -- Weiss and Monneau type monotonicity formulas -- Structure of the singular set -- Estimates in Gaussian spaces: Proofs -- Parabolic Whitney’s extension theorem
  • Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Regularity of geodesic foliations -- 2.1. Transport rays -- 2.2. Whitney's extension theorem for ^{1,1} -- 2.3. Riemann normal coordinates -- 2.4. Proof of the regularity theorem -- Chapter 3. Conditioning a measure with respect to a geodesic foliation -- 3.1. Geodesics emanating from a ^{1,1}-hypersurface -- 3.2. Decomposition into ray clusters -- 3.3. Needles and Ricci curvature -- Chapter 4. The Monge-Kantorovich problem -- Chapter 5. Some applications -- 5.1. The inequalities of Buser, Ledoux and E. Milman -- 5.2. A Poincaré inequality for geodesically-convex domains -- 5.3. The isoperimetric inequality and its relatives -- Chapter 6. Further research -- Appendix: The Feldman-McCann proof of Lemma 2.4.1 -- Bibliography -- Back Cover