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What is a Quadrature Domain?

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Quadrature Domains and Their Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 156))

Abstract

We give an overview of the theory of quadrature domains with indications of some if its ramifications.

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References

  1. D. Aharonov, H.S. Shapiro, A minimal-area problem in conformal mapping — preliminary report, Research bulletin TRITA-MAT-1973-7, Royal Institute of Technology, 34 pp.

    Google Scholar 

  2. D. Aharonov, H.S. Shapiro, Domains in which analytic functions satisfy quadrature identities, J. Analyse Math. 30 (1976), 39–73.

    MathSciNet  Google Scholar 

  3. D. Aharonov, H.S. Shapiro, A minimal-area problem in conformal mapping — preliminary report: Part II, Research bulletin TRITA-MAT-1978-5, Royal Institute of Technology, 70 pp.

    Google Scholar 

  4. D. Aharonov, H.S. Shapiro, A. Solynin, A minimal area problem in conformal mapping, J.Analyse Math. 78 (1999), 157–176.

    MathSciNet  Google Scholar 

  5. D. Aharonov, H.S. Shapiro, A. Solynin, A minimal area problem in conformal mapping. II, J.Analyse Math. 83 (2001), 239–259.

    MathSciNet  Google Scholar 

  6. D. Armitage, M. Goldstein, Quadrature and harmonic approximation of subharmonic functions in strips, J. London Math. Soc. 46 (1992), 171–179.

    MathSciNet  Google Scholar 

  7. D. Armitage, S. Gardiner, Best one-sided L1-approximation by harmonic and subharmonic functions, Advances in multivariate approximation (Witten-Bommenholz, 1998), 43–56, Math. Res., 107, Wiley-VCH, Berlin, 1999.

    Google Scholar 

  8. Y. Avci, Quadrature identities and the Schwarz Function, Doctoral disseertation, Stanford, 1977.

    Google Scholar 

  9. Y. Avci, Characterization of shell domains by quadrature identities, J. London Math. Soc. 23 (1980), 123–128.

    MathSciNet  Google Scholar 

  10. F. Bahrami, A. Chademan, Existence of unbounded quadrature domains for the p-Laplace operator, Bull. Iranian Math. Soc. 14, (1998), 623–681.

    MathSciNet  Google Scholar 

  11. F. Bahrami, A. Chademan, Existence of a uniform bound for quadrature domains associated to p-Laplacian, Analysis (Munich) 19, (1999), 319–326.

    MathSciNet  Google Scholar 

  12. S. Bell, Quadrature domains and kernel function zipping preprint 2003.

    Google Scholar 

  13. S. Bell, The Bergman kernel and quadrature domains in the plane, this volume.

    Google Scholar 

  14. L. Bers: An approximation theorem, J. Analyse Math. 14 (1965), 1–4.

    MATH  MathSciNet  Google Scholar 

  15. L.A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta. Math. 139 (1977) 155–184.

    MathSciNet  Google Scholar 

  16. L.A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Diff. Eq. 5 (1980), 427–448.

    MATH  MathSciNet  Google Scholar 

  17. L.A. Caffarelli, L. Karp, H. Shahgholian, Regularity of a free boundary problem with application to the Pompeiu problem, Ann. Math. 151, (2000), 269–292.

    MathSciNet  Google Scholar 

  18. R.W. Carey, J.D. Pincus, An exponential formula for determining functions, Indiana Univ. Math. J. 23 (1974), 1031–1042.

    MathSciNet  Google Scholar 

  19. J. Conway, L. Yang, Some open problems in the theory of subnormal operators, Holomorphic spaces (Berkeley, CA, 1995), 201–209, Math. Sci. Res. Inst. Publ., 33, Cambridge Univ. Press, Cambridge, 1998.

    Google Scholar 

  20. D. Crowdy, Multipolar vortices and algebraic curves, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), 2337–2359.

    MATH  MathSciNet  Google Scholar 

  21. D. Crowdy, Constructing multiply-connected quadrature domains I: algebraic curves, preprint 2003.

    Google Scholar 

  22. D. Crowdy, J. Marshall, Constructing multiply-connected quadrature domains, SIAM J. Appl. Math. 64 (2004), 1334–1359.

    Article  MathSciNet  Google Scholar 

  23. D. Crowdy, Quadrature domains and fluid dynamics, in this volume.

    Google Scholar 

  24. P.J. Davis, Triangle formulas in the complex plane, Math. of Comp. 18 (1964), 569–577.

    MATH  Google Scholar 

  25. P.J. Davis, Double integrals expressed as single integrals or interpolatory functionals, J. Approx. Theory 5 (1972), 276–307.

    Article  MATH  Google Scholar 

  26. P.J. Davis, The Schwarz Function and its Applications, Carus Math. Mongraphs No.17, Math. Assoc. Amer., 1974.

    Google Scholar 

  27. P.J. Davis, H. Pollak, On the analytic continuation of mapping functions, Trans. Amer. Math. Soc. 87 (1958), 198–225.

    MathSciNet  Google Scholar 

  28. E. DiBenedetto, A. Friedman, The illposed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282 (1984) 183–204.

    MathSciNet  Google Scholar 

  29. E. DiBenedetto, A. Friedman, Bubble growth in porous media, Indiana Univ. Math. J. 35 (1986), 573–606.

    MathSciNet  Google Scholar 

  30. P. Duren, D. Khavinson, H.S. Shapiro, C. Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J. Math. 157 (1993), 37–56.

    MathSciNet  Google Scholar 

  31. P. Duren, A. Schuster, Bergman spaces, Math. Surveys and Monographs vol. 100, Amer. Math. Soc., 2004.

    Google Scholar 

  32. P. Ebenfelt, Singularities encountered by the analytic continuation of solutions to Dirichlet’s problem, Complex Variables 20 (1992), 75–91.

    MATH  MathSciNet  Google Scholar 

  33. P. Ebenfelt, Analytic continuation of certain domain functions in quadrature domains and domains bounded by ellipses, Complex Variables 22 (1993), 69–83.

    MATH  MathSciNet  Google Scholar 

  34. P. Ebenfelt, Singularities of the solution to a certain Cauchy problem and applications to the Pompeiu problem, Duke Math. J. 71 (1993), 119–142.

    Article  MATH  MathSciNet  Google Scholar 

  35. C.M. Elliott, J.R. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Pitman, London (1982).

    Google Scholar 

  36. B. Epstein, On the mean-value property of harmonic functions, Proc. Amer. Math. Soc. 13 (1962), 830.

    MATH  MathSciNet  Google Scholar 

  37. B. Epstein, M. Schiffer, On the mean-value property of harmonic functions, J. Analyse Math. 14 (1965), 109–111.

    MathSciNet  Google Scholar 

  38. N. Feldman, Subnormal operators, self-commutators and pseudocontinuations, Integral Equations Operator Theory 37, (2000), 402–422.

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Friedman, Variational Principles and Free Boundaries, Wiley and Sons, 1982.

    Google Scholar 

  40. A. Friedman, M. Sakai, A characterization of null quadrature domains in Rn, Indiana Univ. Math. J. 35 (1986), 607–610.

    Article  MathSciNet  Google Scholar 

  41. L.A. Galin, Unsteady seepage with a free surface, Dokl. Akad. Nauk SSSR 47 (1945), 250–253. (Russian)

    MathSciNet  Google Scholar 

  42. K.A. Gillow and S.D. Howison, A bibliography of free and moving boundary problems for Hele-Shaw and Stokes flow, published electronically at URL http://www.maths.ox.ac.uk/~howison/Hele-Shaw.

    Google Scholar 

  43. P.G.W. Glare (ed.) Oxford Latin Dictionary, Clarendon Press, Oxford, 1982.

    Google Scholar 

  44. B. Gustafsson, Quadrature identities and the Schottky double, Acta Appl. Math. 1 (1983), 209–240.

    Article  MATH  MathSciNet  Google Scholar 

  45. B. Gustafsson, Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows, SIAM J. Math. Anal. 16 (1985), 279–300.

    Article  MATH  MathSciNet  Google Scholar 

  46. B. Gustafsson: Applications of half-order differentials on Riemann surfaces to quadrature identities for arc-length, J. Analyse Math. 49 (1987), 54–89.

    MATH  MathSciNet  Google Scholar 

  47. B. Gustafsson, Singular and special points on quadrature domains from an algebraic geometric point of view, J. Analyse Math. 51 (1988), 91–117.

    MATH  MathSciNet  Google Scholar 

  48. B. Gustafsson, On quadrature domains and an inverse problem in potential theory, J. Analyse Math. 55 (1990), 172–216.

    MATH  MathSciNet  Google Scholar 

  49. B. Gustafsson: A distortion theorem for quadrature domains for harmonic functions, J. Math. Anal. Appl. 202 (1996), 169–182.

    Article  MATH  MathSciNet  Google Scholar 

  50. B. Gustafsson, On mother bodies of convex polyhedra, SIAM J. Math.Anal. 29:5 (1998), 1106–1117.

    Article  MATH  MathSciNet  Google Scholar 

  51. B. Gustafsson, Lectures on Balayage, preprint 2003.

    Google Scholar 

  52. B. Gustafsson, M. Putinar, An exponential transform and regularity of free boundaries in two dimensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 26 (1998), 507–543.

    MathSciNet  Google Scholar 

  53. B. Gustafsson, M. Putinar, Linear analysis of quadrature domains II, Israel J.Math. 119 (2000), 187–216.

    MathSciNet  Google Scholar 

  54. B. Gustafsson, M. Putinar, On exact quadrature formulas for harmonic functions on polyhedra, Proc. Amer. Math. Soc. 128 (1999), 1427–1432.

    MathSciNet  Google Scholar 

  55. B. Gustafsson, M. Putinar, Linear analysis of quadrature domains IV, in this volume.

    Google Scholar 

  56. B. Gustafsson, C. He, P. Milanfar, M. Putinar, Reconstructing planar domains from their moments, Inverse Problems 16 (2000), 1053–1070.

    Article  MathSciNet  Google Scholar 

  57. B. Gustafsson, M. Sakai, Properties of some balayage operators with applications to quadrature domains and moving boundary problems, Nonlinear Anal. 22 (1994), 1221–1245.

    Article  MathSciNet  Google Scholar 

  58. B. Gustafsson, M. Sakai, On potential theoretic skeletons of polyhedra, Geom. Didicata 76 (1999), 1–30.

    MathSciNet  Google Scholar 

  59. B. Gustafsson, M. Sakai, Sharp estimates of the curvature of some free boundaries in two dimensions, Ann. Acad. Sci. Fenn. Math. 28 (2003), 123–142.

    MathSciNet  Google Scholar 

  60. B. Gustafsson, M. Sakai, On the curvature of the free boundary for the obstacle problem in two dimensions, Monatshefte für Mathematik, 142 (2004), 1–5.

    Article  MathSciNet  Google Scholar 

  61. B. Gustafsson, H. Shahgholian, Existence and geometric properties of solutions of a free boundary problem in potential theory, J. Reine Angew. Math. 473 (1996), 137–179.

    MathSciNet  Google Scholar 

  62. B. Gustafsson, M. Sakai, H.S. Shapiro: On domains in which harmonic functions satisfy generalized mean-value properties, Potential Anal. 7 (1997), 467–484.

    Article  MathSciNet  Google Scholar 

  63. B. Gustafsson, A. Vasilév, Complex and Potential Analysis in Hele-Shaw cells, Lecture notes, in preparation, 2004.

    Google Scholar 

  64. V.P. Havin, Approximation in the mean by analytic functions, Dokl. Akad. Nauk. SSSR 178, pp. 1025–1028 (Russion; Engl. transl. in Siviet Math. Dokl. 9 (1968), 245–258.

    MATH  MathSciNet  Google Scholar 

  65. H. Hedenmalm, A factorization theorem for square area-integrable analytic functions, J. Reine Angew. Math. 422 (1991), 45–68.

    MATH  MathSciNet  Google Scholar 

  66. H. Hedenmalm, Recent progress and open problems in the Bergman space, in this volume.

    Google Scholar 

  67. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, 199, Springer-Verlag, New York, 2000.

    Google Scholar 

  68. H. Hedenmalm, S. Shimorin, Hele-Shaw flow on hyperbolic surfaces, J.Math Pures Appl. 81 (2002), 187–222.

    MathSciNet  Google Scholar 

  69. H.S. Hele-Shaw, The flow of water, Nature 58 (1898), 34–36.

    Google Scholar 

  70. H.S. Hele-Shaw, On the motion of a viscous fluid between two parallel plates, Trans. Royal Inst. Nav. Archit., London 40 (1898), 218.

    Google Scholar 

  71. A. Henrot, Subsolutions and supersolutions in free boundary problems Ark. Mat. 32 (1994), 79–98.

    MATH  MathSciNet  Google Scholar 

  72. G. Herglotz, Über die analytische Fortsetzung des Potentials ins Innere der anziehenden Massen, Gekrönte Preisschr. der Jablonowskischen Gesellsch. zu Leipzig, 56 pp., Reproduced in ‘Gustav Herglotz-Gesammelte Schriften’, Vandenhoeck & Ruprecht, Göttingen, 1979, pp. 299–355.

    Google Scholar 

  73. Y. Hohlov, S.D. Howison, C. Huntingford, J.R. Ockendon, A.A. Lacey, A model for non-smooth free boundaries in Hele-Shaw flows, Quart. J. Appl. Math. 47 (1994), 107–128.

    MathSciNet  Google Scholar 

  74. S.D. Howison, Bubble growth in porous media and Hele Shaw flow, Proc. Roy. Soc. Edinburgh A102 (1985), 141–148.

    MathSciNet  Google Scholar 

  75. V. Isakov, Inverse Source Problems, AMS Math. Surveys and Monographs 34, Providence Rhode Island, 1990.

    Google Scholar 

  76. G. Johnsson, The Cauchy problem in CN for linear second order partial differential equations with data on a quadric surface, Trans. Amer. Math. Soc. 344 (1994), 1–48.

    MATH  MathSciNet  Google Scholar 

  77. G. Johnsson, Global existence results for linear partial differential equations, J. Differential Equations 115 (1995), 416–440.

    Article  MATH  MathSciNet  Google Scholar 

  78. L. Karp, Construction of quadrature domains in Rnfrom quadrature domains in R2, Complex variables 17 (1992), 179–188.

    MathSciNet  Google Scholar 

  79. L. Karp, Generalized Newtonian potential and its applications, J. Math. Anal. Appl. 174 (1993), 480–497.

    Article  MATH  MathSciNet  Google Scholar 

  80. L. Karp, On the Newtonian potential of ellipsoids, Complex Variables 25 (1995), 367–372.

    MathSciNet  Google Scholar 

  81. L. Karp, Multivalued analytic continuation of the Cauchy transform, preprint 2003.

    Google Scholar 

  82. L. Karp, A. Margulis, Newtonian potential theory for unbounded sources and applications to free boundary problems, J. Analyse Math. 70 (1996), 1–63.

    MathSciNet  Google Scholar 

  83. L. Karp, H. Shahgholian, Regularity of a free boundary problem, J. Geom. Anal. 9, (1999), 653–669.

    MathSciNet  Google Scholar 

  84. D. Khavinson, Singularities of harmonic functions in Cn, Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), 207–217, Proc. Sympos. Pure Math., 52, Part 3, Amer. Math. Soc., Providence, RI, 1991.

    Google Scholar 

  85. D. Khavinson, Holomorphic Partial Differential Equations and Classical Potential Theory, Departamento de Análisis Matemático, Universidad de La Laguna, Tenerife, 1996.

    Google Scholar 

  86. D. Khavinson, H.S. Shapiro, The Schwarz potential in Rnand Cauchy’s problem for the Laplace equation, Research bulletin TRITA-MAT-1986-36, Royal Institute of Thechnology, 112 pp.

    Google Scholar 

  87. D. Khavinson, H.S. Shapiro, The Vekua hull of a plane domain, Complex Variables 14 (1990), 117–128.

    MathSciNet  Google Scholar 

  88. D. Kinderlehrer, L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 373–391.

    MathSciNet  Google Scholar 

  89. D. Kinderlehrer, G. Stampacchia, Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.

    Google Scholar 

  90. O.I. Kounchev, The partial balayage materic bodies and optimization problems in gravimetry, in Inverse Modeling in Exploration Geophysics, Proceedings of the 6th International Mathematical Geophysics Seminar in Berlin, Feb. 3–6, 1988 (eds. A. Vogel, R. Gorenflo, B. Kummer, C.O. Ofoegbu), Vieweg & Sohn, Braunschweig/Wiesbaden.

    Google Scholar 

  91. O.I. Kounchev, Obtaining materic bodies through concentration and optimization of a linear functional, in Geophysical Data Inversion Methods and Applications, Proceedings of the 7th International Mathematical Geophysics Seminar in Berlin, Feb. 8–11, 1989 (eds. A. Vogel, R. Gorenflo, C.O. Ofoegbu, B. Ursin), Vieweg & Sohn, Braunschweig/Wiesbaden.

    Google Scholar 

  92. J. Leray, Uniformisation de la solution du problème linéaire analyticque de Cauchy, près de la varieté qui porte les données de Cauchy, Bull. Soc. Math. France 85, 389–430.

    Google Scholar 

  93. A.A. Levin, An example of a doubly connected domain which admits a quadrature identity, Proc. Amer. Math. Soc. 60 (1976), 163–168.

    MathSciNet  Google Scholar 

  94. A.S. Margulis, The moving boundary problem of potential theory, Adv. Math. Sci. Appl. 5(2) (1995), 603–629.

    MATH  MathSciNet  Google Scholar 

  95. R. Marrero, I. Maria, Analytic characterization of annuli, Rev. Acad. Canaria Cienc. 1 (1990), 147–153.

    MathSciNet  Google Scholar 

  96. M. Martin, M. Putinar, Lectures on Hyponormal Operators, Birkhäuser, Basel, 1989.

    Google Scholar 

  97. J. McCarthy, L. Yang, Cyclic subnormal operators with finite-rank self-commutators, Proc. Roy. Irish Acad. Sect. A 95 (1995), 173–177.

    MathSciNet  Google Scholar 

  98. J. McCarthy, L. Yang, Subnormal operators and quadrature domains, Adv. Math. 127 (1997), 52–72.

    Article  MathSciNet  Google Scholar 

  99. N.S. Nadirashvili, Universal classes of uniqueness of domains in the inverse problem of Newtonian potential theory, Soviet Math. Dokl. 44 (1992), 287–290.

    MathSciNet  Google Scholar 

  100. C. Neumann, Über das logarithmische Potential einer gewissen Ovalfläche, Abh. der math.-phys. Klasse der Königl. Sächs. Gesellsch. der Wiss. zu Leibzig 59 (1907), pp. 278–312.

    MATH  Google Scholar 

  101. C. Neumann, Über das logarithmische Potential einer gewissen Ovalfläche, Zweite Mitteilung, ibib. vol. 60 (1908), pp. 53–56. Dritte Mitteilung, ibid. pp. 240–247.

    MATH  Google Scholar 

  102. M. Putinar, On a class of finitely determined planar domains, Math. Res. Lett. 1 (1994), 389–398.

    MATH  MathSciNet  Google Scholar 

  103. M. Putinar, Linear analysis of quadrature domains, Ark. Mat. 33 (1995), 357–376.

    MATH  MathSciNet  Google Scholar 

  104. M. Putinar, Extremal solutions of the two-dimensional L-problem of moments, J. Funct.An. 136 (1996), 331–364.

    MATH  MathSciNet  Google Scholar 

  105. M. Putinar, Extremal solutions of the two-dimensional L-problem of moments II, J.Approx.Th. 92 (1998), 38–58.

    Article  MATH  MathSciNet  Google Scholar 

  106. M. Putinar, Matrix analysis and the Friedrichs operator of a quadrature domain, Linear Algebra Appl. 270 (1998), 215–229.

    MATH  MathSciNet  Google Scholar 

  107. M. Putinar, Linear analysis of quadrature domains III, Math. Anal. Appl. 239 (1999), 101–117.

    Article  MATH  MathSciNet  Google Scholar 

  108. M. Putinar, A renormalized Riesz transform and applications, Advances in Constructive Approximation, to appear.

    Google Scholar 

  109. G. Putinar, M. Putinar, Root separation on generalized lemniscates, Hokkaido Math. J. 30 (2001), 705–716.

    MathSciNet  Google Scholar 

  110. M. Putinar, H.S. Shapiro, The Friedrichs operator of a planar domain, Complex analysis, operators, and related topics, 303–330, Oper. Theory Adv. Appl. 113, Birkhäuser, Basel, 2000.

    Google Scholar 

  111. M. Putinar, H.S. Shapiro, The Friedrichs operator of a planar domain II, Recent advances in operator theory and related topics (Szeged, 1999), 519–551, Oper. Theory Adv. Appl. 127, Birkhäuser, Basel, 2001.

    Google Scholar 

  112. 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.

    MATH  Google Scholar 

  113. S. Richardson, Hele-shaw flows with time-dependent free boundaries involving a multiply connected fluid region, European J. Appl. Math. 12 (2001), 571–599.

    MATH  MathSciNet  Google Scholar 

  114. J.F. Rodrigues, Obstacle problems in Mathematical Physics, North-Holland, Amsterdam 1987.

    Google Scholar 

  115. P.G. Saffman, F.P. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid, Proc. Royal Soc. London Ser. A 245 (1958), 312–329.

    MathSciNet  Google Scholar 

  116. M. Sakai, On basic domains of extremal functions, Kodai Math. Sem. Rep. 24 (1972), 251–258.

    MATH  MathSciNet  Google Scholar 

  117. M. Sakai, A moment problem on Jordan domains, Proc. Amer.Math. Soc. 70 (1978), 35–38.

    MATH  MathSciNet  Google Scholar 

  118. M. Sakai, Analytic functions with finite Dirichlet integrals on Riemann surfaces, Acta Math. 142 (1979), 199–220.

    MATH  MathSciNet  Google Scholar 

  119. M. Sakai, The submeanvalue property of subharmonic functions and its application to the estimation of the Gaussian curvature of the span metric, Hiroshima Math. J. 9 (1979), 555–593

    MATH  MathSciNet  Google Scholar 

  120. M. Sakai, Null quadrature domains, J. Analyse Math. 40 (1981), 144–154.

    MATH  MathSciNet  Google Scholar 

  121. M. Sakai, Quadrature Domains, Lect. Notes Math. 934, Springer-Verlag, Berlin-Heidelberg 1982.

    Google Scholar 

  122. M. Sakai, Application of variational inequalities to the existence theorem on quadrature domains, Trans. Amer. Math. Soc. 276 (1983), 267–279.

    MATH  MathSciNet  Google Scholar 

  123. M. Sakai, Solutions to the obstacle problem as Green potentials, J. Analyse Math. 44 (1984/85) 97–116.

    MathSciNet  Google Scholar 

  124. M. Sakai, Domains having null complex moments, Complex Variables 7 (1987), 313–319.

    MATH  Google Scholar 

  125. M. Sakai, Finiteness of the family of simply connected quadrature domains, pp. 295–305 in “Potential Theory” (editors J. Kral, I. Netuka, J. Vesely), Plenum Publishing Corporation, 1988.

    Google Scholar 

  126. M. Sakai, An index theorem on singular points and cusps of quadrature doamins, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 119–131, Math. Res. Inst. Publ., 10, Springer, New York, 1988.

    Google Scholar 

  127. M. Sakai, Regularity of boundary having a Schwarz function, Acta Math. 166 (1991), 263–297.

    MATH  MathSciNet  Google Scholar 

  128. M. Sakai, Regularity of free boundaries in two dimensions, Ann. Scuola Norm Sup. Pisa Cl. Sci. (4) 20 (1993), 323–339.

    MATH  MathSciNet  Google Scholar 

  129. M. Sakai, Regularity of boundaries of quadrature domains in two dimensions, SIAM J. Math. Anal. 24 (1994), 341–364.

    Google Scholar 

  130. M. Sakai, Sharp estimates of the distance from a flxed point to the frontier of Hele-Shaw flow, Potential Anal. 8 (1998), 277–302.

    Article  MATH  MathSciNet  Google Scholar 

  131. M. Sakai, Linear combinations of harmonic measures and quadrature domains of signed measures with small supports, Proc. Edinburgh Math. Soc. (2) 42 (1999),433–444.

    MATH  MathSciNet  Google Scholar 

  132. T. Savina, B. Sternin, V. Shatalov, Notes on “mother body” problem in geographics, preprint 1995.

    Google Scholar 

  133. F. Schottky Über die conforme Abbildung mehrfach zusammenhängender ebener Flächen, Crelles Journal 83 (1877), 300–351.

    MATH  Google Scholar 

  134. M. Schiffer, N.S. Hawley, Half-order differentials on Riemann surfaces, ActaMath. 115 (1966), 199–236.

    MathSciNet  Google Scholar 

  135. M. Schiffer, D.C. Spencer, Functionals of Finite Riemann Surfaces, Princeton Univ. Press, Princeton, 1954.

    Google Scholar 

  136. H. Shahgholian, Unbounded quadrature domains in Rn (n ≥ 3), J.Analyse Math. 57 (1991), 281–291, 297–298.

    Google Scholar 

  137. H. Shahgholian, On quadrature domains and the Schwarz potential, J. Math. Anal. Appl. 171 (1992), 61–78.

    Article  MATH  MathSciNet  Google Scholar 

  138. H. Shahgholian, A characterization of the sphere in terms of single-layer potentials, Proc. Amer. Math. Soc. 115 (1992), 1167–1168.

    MATH  MathSciNet  Google Scholar 

  139. H. Shahgholian, Convexity and uniqueness in an inverse problem of potential theory, Proc. Amer. Math. Soc. 116 (1992), 1097–1100.

    MATH  MathSciNet  Google Scholar 

  140. H. Shahgholian, On the Newtonian potential of heterogeneous ellipsoids, SIAM J. Math. Anal. 24 (1993), 341–364.

    MathSciNet  Google Scholar 

  141. H. Shahgholian, Quadrature surfaces as free boundaries Ark. Mat. 32 (1994), 475–492.

    MATH  MathSciNet  Google Scholar 

  142. H. Shahgholian, Existence of quadrature surfaces for positive measures with finite support Potential Analysis 3 (1994), 245–255.

    Article  MATH  MathSciNet  Google Scholar 

  143. H.S. Shapiro, Domains allowing exact quadrature identities for harmonic functions — an approach based on p.d.e., Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983), 335–354, Internat. Schriftenreihe Numer. Math., 65, Birkhäuser, Basel-Boston, 1984.

    Google Scholar 

  144. H.S. Shapiro, Global aspects of the Cauchy’s problem for the Laplace operator, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989), 309–324, Sem. Conf., 8, EditEl, Rende, 1991.

    MATH  Google Scholar 

  145. H.S. Shapiro, Unbounded quadrature domains, Complex analysis, I (College Park, Md., 1985–86), 287–331, Lecture Notes in Math., 1275, Springer-Verlag, Berlin 1987.

    Google Scholar 

  146. H.S. Shapiro, The Schwarz function and its generalization to higher dimensions, Uni. of Arkansas Lect. Notes Math. Vol. 9, Wiley, New York, 1992.

    Google Scholar 

  147. H.S. Shapiro, Quasi-balayage and a priori estimates for the Laplace operator, Multivariate approximation (Witten-Bommerholz 1996), 203–230, 231–254, Math. Res. 101, Akademie Verlag, Berlin, 1997.

    Google Scholar 

  148. H.S. Shapiro, C. Ullemar, Conformal mappings satisfying certain extremal properties, and associated quadrature identities, Research bulletin TRITA-MAT-1981-6, Royal Institute of Technology, 40 pp.

    Google Scholar 

  149. T. Sjödin, Mother body for the ellipse, manuscript.

    Google Scholar 

  150. T. Sjödin, Quadrature identities and deformation of quadrature domains, this volume.

    Google Scholar 

  151. B. Sternin, V. Shatalov, On the problem of balayage in the space Rn, (Russian) Mat. Zametki 54 (1993), 90–112, 160; translation in Math. Notes 54 (1993), 1246–1260.

    MathSciNet  Google Scholar 

  152. S.A. Stewart, D. Xia, A class of subnormal operators with finite rank self-commutators, Integral Equations Operator Theory 44 (2002), 370–382.

    Article  MathSciNet  Google Scholar 

  153. F.R. Tian, A Cauchy integral approach to Hele-Shaw flow problems with a free boundary: the case of zero surface tension, Arch. Rational Mech. Anal. 135 (1996), 175–195.

    Article  MATH  MathSciNet  Google Scholar 

  154. C. Ullemar, Symmetric plane domains satisfying two-point quadrature identities for analytic functions, Research bulletin TRITA-MAT-1977-24, Royal Institute of Technology, 23 pp.

    Google Scholar 

  155. C. Ullemar, A uniqueness theorem for domains satisfying a quadrature identity for analytic functions, Research bulletin TRITA-MAT-1980-37, Royal Institute of Technology, 59 pp.

    Google Scholar 

  156. A.N. Varchenko, P.I. Etingof, Why the Boundary of a Round Drop Becomes a Curve of Order Four, American Mathematical Society AMS University Lecture Series, Volume 3, Providence, Rhode Island 1992.

    Google Scholar 

  157. D. Xia, Hyponormal operators with finite rank self-commutators and quadrature domains, J. Math. Anal. Appl. 203 (1996), 540–559.

    Article  MATH  MathSciNet  Google Scholar 

  158. D. Xia, On pure subnormal operators with finite rank self-commutators and related operator-tuples, Integral Equations Operator Theory 24 (1996), 106–125.

    Article  MATH  MathSciNet  Google Scholar 

  159. D. Xia, Trace formulas for some operators related to quadrature domains in Riemann surfaces, Integral Equations Operator Theory 47 (2003), 123–130.

    Article  MATH  MathSciNet  Google Scholar 

  160. D.V. Yakubovich, Subnormal operators of finite type, II Structure theorems, Rev. Mat. Iberoamericana 14 (1998), 623–681.

    MATH  MathSciNet  Google Scholar 

  161. D.V. Yakubovich, A note on hyponormal operators and associated with quadrature domains, Operator theory, system theory and related topics, Beer-Sheva/Rehovot, 1997), 513–525, Oper. Theory Adv. Appl., 123, Birkhäuser, Basel, 2001.

    Google Scholar 

  162. L. Zalcman, Some inverse problems of potential theory, Contemp. Math. 63 (1987), 337–350.

    MATH  MathSciNet  Google Scholar 

  163. D. Zidarov, Inverse Gravimetric Problem in Geoprospecting and Geodesy, Elsevier, Amsterdam, 1990. (First edition 1968, in Russian.).

    Google Scholar 

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Gustafsson, B., Shapiro, H.S. (2005). What is a Quadrature Domain?. In: Ebenfelt, P., Gustafsson, B., Khavinson, D., Putinar, M. (eds) Quadrature Domains and Their Applications. Operator Theory: Advances and Applications, vol 156. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7316-4_1

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