×

On quadrature domains and an inverse problem in potential theory. (English) Zbl 0745.31002

The author considers relations between quadrature domains and the inverse problem in potential theory which are quite close and deep. He introduces some order in the class of domains determined by comparison of their potentials, studies extremal domains with respect to this order and gives some interesting examples. About the inverse problem of potential theory we refer to the recent book of the reviewer [Inverse source problems (Providence 1990; Zbl 0721.31002). There are many challenging questions there.
Reviewer: V.Isakov (Wichita)

MSC:

31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35R30 Inverse problems for PDEs

Citations:

Zbl 0721.31002
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] D. Aharonov, M. M. Schiffer and L. Zalcman,Potato Kugel, Isr. J. Math.40 (1981), 331–339. · Zbl 0496.31006
[2] D. Aharonov and H. S. Shapiro,Domains on which analytic functions satisfy quadrature identities, J. Analyse Math.30 (1976), 39–73. · Zbl 0337.30029
[3] A. Anacona,Continuité des contractions dans les espaces de Dirichlet, C.R. Acad. Sci. Paris282 (1976), 871–873.
[4] Y. Avci,Quadrature identities and the Schwarz function, Doctoral dissertation, Stanford University, 1977.
[5] C. Baiocchi and A. Capelo,Variational and Quasi-variational Inequalities. Applications to Free Boundary Problems, Wiley, Chichester, 1984. · Zbl 0551.49007
[6] S. Bergman,Integral Operators in the Theory of Linear Partial Differential Equations, Springer-Verlag, Berlin, 1961. · Zbl 0093.28701
[7] L. Bers,An approximation theorem, J. Analyse Math.14 (1965), 1–4. · Zbl 0134.05304
[8] H. Brezis and F. E. Browder,Some properties of higher order Sobolev spaces, J. Math. Pures Appl.61 (1982), 245–259. · Zbl 0512.46034
[9] L. A. Caffarelli,Compactness methods in free boundary problems, Commun. Partial Diff. Equ.5 (1980), 427–448. · Zbl 0437.35070
[10] P. Davis,The Schwarz Function and its Application, Carus Mathematical Monographs, 1974.
[11] B. Epstein and M. Schiffer,On the mean-value property of harmonic functions, J. Analyse Math.14 (1965), 109–111. · Zbl 0131.10003
[12] A. Friedman,Variational Principles and Free Boundary Problems, Wiley, New York, 1982. · Zbl 0564.49002
[13] B. Gustafsson, TRITA-MAT-1981-9; the first version of [Gu 3].
[14] B. Gustafsson,Quadrature identities and the Schottky double, Acta Appl. Math.1 (1983), 209–240. · Zbl 0559.30039
[15] B. Gustafsson,Applications of variational inequalities to a moving boundary problem for Hele Shaw flows, SIAM J. Math. Anal.16 (1985), 279–300. · Zbl 0605.76043
[16] B. Gustafsson,Existence of weak backward solutions to a generalized Hele Shaw flow moving boundary problem, Nonlinear Anal.9 (1985), 203–215. · Zbl 0556.35130
[17] B. Gustafsson,An ill-posed moving boundary problem for doubly-connected domains, Ark. Mat.25 (1987), 231–253. · Zbl 0635.35085
[18] B. Gustafsson,Singular and special points on quadrature domains from an algebraic geometric point of view, J. Analyse Math.51 (1988), 91–117. · Zbl 0656.30034
[19] N. M. Günter,Potential Theory and its Applications to Basic Problems of Mathematical Physics, Frederick Ungar Publ. Co., New York, 1967.
[20] L. L. Helms,Introduction to Potential Theory, Wiley, New York, 1969. · Zbl 0188.17203
[21] E. Hopf,A remark on linear elliptic differential equations of second order, Proc. Am. Math. Soc.3 (1952), 791–793. · Zbl 0048.07802
[22] V. Isakov,Inverse source problems, to appear. · Zbl 0721.31002
[23] D. Kinderlehrer and L. Nirenberg,Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)4 (1977), 373–391. · Zbl 0352.35023
[24] D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980. · Zbl 0457.35001
[25] U. Küran,On the mean-value property of harmonic functions, Bull. London Math. Soc.4 (1972), 311–312. · Zbl 0257.31006
[26] J.-F. Rodringuez,Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987.
[27] M. Sakai,A moment problem on Jordan domains, Proc. Am. Math. Soc.70 (1978), 35–38. · Zbl 0394.30027
[28] M. Sakai,Quadrature Domains, Lecture Notes in Math., Vol. 934, Springer-Verlag, Berlin, 1982. · Zbl 0483.30001
[29] M. Sakai,Applications of variational inequalities to the existence theorem on quadrature domains, Trans. Am. Math. Soc.276 (1983), 267–279. · Zbl 0515.31001
[30] M. Sakai,Solutions to the obstacle problem as Green potentials, J. Analyse Math.44 (1984/85), 97–116. · Zbl 0577.49005
[31] M. Sakai,The obstacle problem and its application, Research Institute for Mathematical Sciences, Kokyuroku No. 502, Kyoto University, 1983, pp. 1–12.
[32] M. Sakai,Domains having null complex moments, Complex Variables7 (1987), 313–319. · Zbl 0555.30022
[33] M. Sakai,An index theorem on singular points and cusps of quadrature domains, inHolomorphic Functions and Moduli, Vol. I, D. Drasin (ed.), Springer-Verlag, New York, 1988, pp. 119–131.
[34] M. Sakai,Finiteness of the family of simply connected quadrature domains, inPotential Theory, Proceedings of a conference on potential theory, July 19–24, 1987 in Prague, J. Kral, J. Lukes, I. Netuka, and J. Vesely (eds.), Plenum Press, New York, 1988, pp. 295–305.
[35] M. Sakai,Regularity of a boundary having a Schwarz function, Acta Math., to appear. · Zbl 0728.30007
[36] H. S. Shapiro,Domains allowing exact quadrature identities for harmonic functions –an approach based on PDE, inAnniversary Volume on Approximation Theory and Functional Analysis, P. L. Butzer, R. L. Stens and B. Sz.-Nagy (eds.), ISNM 65, BirkhÄuser-Verlag, Basel, Boston, Stuttgart, 1984.
[37] H. Shapiro,Unbounded quadrature domains, inComplex Analysis I, Proceedings, Univ. of Maryland 1085–86, C. A. Berenstein (ed.), Lecture Notes in Mathematics, Vol. 1275, Springer-Verlag, Berlin, 1987, pp. 287–331.
[38] F. Treves,Basic Linear Partial Differential Equations, Academic Press, New York, 1975.
[39] L. Zalcman,Some inverse problems of potential theory, Contemp. Math. 63 (1987), 337–350. · Zbl 0641.31003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.