Duke Mathematical Journal

A minimization problem with free boundary related to a cooperative system

Luis A. Caffarelli, Henrik Shahgholian, and Karen Yeressian

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Abstract

We study the minimum problem for the functional Ω(|u|2+Q2χ{|u|>0})dx with the constraint ui0 for i=1,,m, where ΩRn is a bounded domain and u=(u1,,um)H1(Ω;Rm). First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set {|u|>0} is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary Ω{|u|>0}.

Article information

Source
Duke Math. J., Volume 167, Number 10 (2018), 1825-1882.

Dates
Received: 1 November 2016
Revised: 13 January 2018
First available in Project Euclid: 5 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1528185619

Digital Object Identifier
doi:10.1215/00127094-2018-0007

Mathematical Reviews number (MathSciNet)
MR3827812

Zentralblatt MATH identifier
06928112

Subjects
Primary: 35R35: Free boundary problems
Secondary: 35J60: Nonlinear elliptic equations

Keywords
minimization Bernoulli-type free boundary free boundary system regularity

Citation

Caffarelli, Luis A.; Shahgholian, Henrik; Yeressian, Karen. A minimization problem with free boundary related to a cooperative system. Duke Math. J. 167 (2018), no. 10, 1825--1882. doi:10.1215/00127094-2018-0007. https://projecteuclid.org/euclid.dmj/1528185619


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References

  • [1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623–727.
  • [2] N. E. Aguilera, L. A. Caffarelli, and J. Spruck, An optimization problem in heat conduction, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (1987), 355–387.
  • [3] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144.
  • [4] H. W. Alt, L. A. Caffarelli, and A. Friedman, A free boundary problem for quasilinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (1984), 1–44.
  • [5] H. W. Alt, L. A. Caffarelli, and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), no. 2, 431–461.
  • [6] J. Andersson, Optimal regularity for the Signorini problem and its free boundary, Invent. Math. 204 (2016), 1–82.
  • [7] J. Andersson, H. Shahgholian, N. N. Uraltseva, and G. S. Weiss, Equilibrium points of a singular cooperative system with free boundary, Adv. Math. 280 (2015), 743–771.
  • [8] L. A. Caffarelli, D. S. Jerison, and C. E. Kenig, “Global energy minimizers for free boundary problems and full regularity in three dimensions” in Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, Contemp. Math. 350, Amer. Math. Soc., Providence, 2004, 83–97.
  • [9] L. A. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc. 21 (2008), 847–862.
  • [10] M. Conti, S. Terracini, and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math. 195 (2005), 524–560.
  • [11] D. De Silva and D. S. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math. 635 (2009), 1–21.
  • [12] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised ed., Textb. Math., CRC Press, Boca Raton, Fla., 2015.
  • [13] D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math. 46 (1982), 80–147.
  • [14] D. S. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal. 25 (2015), 1240–1257.
  • [15] H. Jiang and F. Lin, A new type of free boundary problem with volume constraint, Comm. Partial Differential Equations 29 (2004), 821–865.
  • [16] C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math. 83, Amer. Math. Soc., Providence, 1994.
  • [17] D. Kinderlehrer, L. Nirenberg, and J. Spruck, Regularity in elliptic free boundary problems, J. Anal. Math. 34 (1978), 86–119.
  • [18] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss. 130, Springer, New York, 1966.
  • [19] G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9 (1999), 317–326.