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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
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.
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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), no. 1-2, 93–107. MR 611303
(82d:76031), http://dx.doi.org/10.1017/S0308210500017315
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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.
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Lars
Inge Hedberg, Approximation in the mean by solutions of elliptic
equations, Duke Math. J. 40 (1973), 9–16. MR 0312071
(47 #633)
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David
Kinderlehrer and Guido
Stampacchia, An introduction to variational inequalities and their
applications, Pure and Applied Mathematics, vol. 88, Academic
Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.
MR 567696
(81g:49013)
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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.
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Makoto
Sakai, Quadrature domains, Lecture Notes in Mathematics,
vol. 934, Springer-Verlag, Berlin-New York, 1982. MR 663007
(84h:41047)
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Makoto
Sakai, Null quadrature domains, J. Analyse Math.
40 (1981), 144–154 (1982). MR 659788
(84e:30069), http://dx.doi.org/10.1007/BF02790159
- [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:
http://dx.doi.org/10.1090/S0002-9947-1983-0684507-2
PII:
S 0002-9947(1983)0684507-2
Keywords:
Quadrature domains,
variational inequalities,
subharmonic functions,
potentials
Article copyright:
© Copyright 1983
American Mathematical Society
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