Applications of variational inequalities to the existence theorem on quadrature domains

Sakai, Makoto

doi:10.1090/S0002-9947-1983-0684507-2

Applications of variational inequalities to the existence theorem on quadrature domains
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Trans. Amer. Math. Soc. 276 (1983), 267-279 Request permission

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

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Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel

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Makoto Sakai, Quadrature domains, Lecture Notes in Mathematics, vol. 934, Springer-Verlag, Berlin-New York, 1982. MR 663007
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