Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Improvement of flatness for vector valued free boundary problems

1 Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA
2 Dipartimento di Matematica, Alma Mater Studiorum Universita di Bologna, Piazzà di Porta San Donato 5, 40126 Bologna, Italy

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: https://www.aimspress.com/newsinfo/1429.html

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies C1,α regularity, as well-known in the scalar case [1, 4]. While in [15] the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of [8]. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in [10, 11].
  Figure/Table
  Supplementary
  Article Metrics

Keywords one-phase free boundary problem; Harnack inequality; vectorial problem; viscosity solution; improvement of flatness

Citation: Daniela De Silva, Giorgio Tortone. Improvement of flatness for vector valued free boundary problems. Mathematics in Engineering, 2020, 2(4): 598-613. doi: 10.3934/mine.2020027

References

  • 1. Caffarelli LA, Alt HW (1981) Existence and regularity for a minimum problem with free boundary. J Reine Angew Math 325: 105-144.
  • 2. Caffarelli LA (1987) A Harnack inequality approach to the regularity of free boundaries. Part I. Lipschitz free boundaries are C1,α. Rev Mat Iberoamericana 3: 139-162.
  • 3. Caffarelli LA (1988) A Harnack inequality approach to the regularity of free boundaries. Part III. Existence theory, compactness, and dependence on x. Ann Scuola Norm Sci 15: 583-602.
  • 4. Caffarelli LA (1989) A Harnack inequality approach to the regularity of free boundaries. Part II. Flat free boundaries are Lipschitz. Commun Pure Appl Math 42: 55-78.
  • 5. Caffarelli LA, Roquejoffre JM, Sire Y (2010) Variational problems for free boundaries for the fractional Laplacian. J Eur Math Soc 12: 1151-1179.
  • 6. Caffarelli LA, Shahgholian H, Yeressian K (2018) A minimization problem with free boundary related to a cooperative system. Duke Math J 167: 1825-1882.    
  • 7. De Philippis G, Spolaor L, Velichkov B (2019) Regularity of the free boundary for the two-phase Bernoulli problem. arXiv:1911.02165.
  • 8. De Silva D (2011) Free boundary regularity for a problem with right hand side. Interface Free Bound 13: 223-238.
  • 9. De Silva D, Ferrari F, Salsa S (2014) On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete Contin Dyn Syst Ser S 7: 673-693.
  • 10. De Silva D, Roquejoffre JM (2012) Regularity in a one-phase free boundary problem for the fractional Laplacian. Ann I H Poincare An 29: 335-367.    
  • 11. De Silva D, Savin O, Sire Y (2014) A one-phase problem for the fractional Laplacian: Regularity of flat free boundaries. Bull Inst Math Acad Sin 9: 111-145.
  • 12. Kriventsov D, Lin FH (2018) Regularity for shape optimizers: The nondegenerate case. Commun Pure Appl Math 71: 1535-1596.    
  • 13. Kriventsov D, Lin FH (2019) Regularity for shape optimizers: The degenerate case. Commun Pure Appl Math 72: 1678-1721.    
  • 14. Mazzoleni D, Terracini S, Velichkov B (2017) Regularity of the optimal sets for some spectral functionals. Geom Funct Anal 27: 373-426.    
  • 15. Mazzoleni D, Terracini S, Velichkov B (2020) Regularity of the free boundary for the vectorial Bernoulli problem. Anal PDE 13: 741-763.    

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved