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Applications of variational inequalities to the existence theorem on quadrature domains


Author: Makoto Sakai
Journal: Trans. Amer. Math. Soc. 276 (1983), 267-279
MSC: Primary 31A05; Secondary 31B05, 49A29
DOI: https://doi.org/10.1090/S0002-9947-1983-0684507-2
MathSciNet review: 684507
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Abstract: In this paper we shall study quadrature domains for the class of subharmonic functions. By using the theory of variational inequalities, we shall give a new proof of the existence and uniqueness theorem. As an application, we deal with Hele-Shaw flows with a free boundary and show that their two weak solutions, one of which was defined by the author using quadrature domains and the other was defined by Gustafsson [3] using variational inequalities, are identical with each other.

References [Enhancements On Off] (What's this?)

  • [1] L. Bers, An approximation theorem, J. Analyse Math. 14 (1965), 1-4. MR 0178287 (31:2545)
  • [2] C. M. Elliott and V. Janovský, A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 93-107. MR 611303 (82d:76031)
  • [3] B. Gustafsson, Applications of variational inequalities to a moving boundary problem for Hele Shaw flows, TRITA-MAT-1981-9, Mathematics, Roy. Inst. Tech., Stockholm, p. 84.
  • [4] L. I. Hedberg, Approximation in the mean by solutions of elliptic equations, Duke Math. J. 40 (1973), 9-16. MR 0312071 (47:633)
  • [5] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980. MR 567696 (81g:49013)
  • [6] S. Richardson, Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972), 609-618.
  • [7] M. Sakai, Quadrature domains, Lecture Notes in Math., vol. 934, Springer-Verlag, Berlin, 1982. MR 663007 (84h:41047)
  • [8] -, Null quadrature domains, J. Analyse Math. 40 (1981), 144-154. MR 659788 (84e:30069)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0684507-2
Keywords: Quadrature domains, variational inequalities, subharmonic functions, potentials
Article copyright: © Copyright 1983 American Mathematical Society

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