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 | References | Similar Articles | Additional Information
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.
- [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