Abstract
We discuss an overdetermined problem in planar multiply connected domains Ω. This problem is solvable in Ω if and only if Ω is a quadrature domain carrying a solid-contour quadrature identity for analytic functions. At the same time the existence of such quadrature identity is equivalent to the solvability of a special boundary value problem for analytic functions. We give a complete solution of the problem in some special cases and discuss some applications concerning the shape of electrified droplets and small air bubbles in a fluid flow.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Aharonov, H. S. Shapiro and A. Yu. Solynin, A minimal area problem in conformal mapping, J. Analyse Math. 78 (1999), 157–176
D. Aharonov, H. S. Shapiro and A. Yu. Solynin, A minimal area problem in conformal mapping II, J. Analyse Math. 83 (2001), 259–288.
G. Alessandrini, A symmetry theorem for condensers, Math. Methods Appl. Sci. 15 (1992) no.5, 315–320.
H. Alexander, Projections of polynomial hulls, J. Functional Analysis 13 (1973), 13–19.
A. Bennet, Symmetry in an overdetermined fourth order elliptic boundary value problem, SIAM J. Math. Anal. 17 (1986), 1354–1358.
R. Berker, Contrainte sur une paroi en contact avec un fluide visqueux classique, un fluide de Stokes, un fluide de Coleman-Noll, C. R. Acad. Sci. Paris 258 (1964), 5144–5147.
Ph. J. Davis, The Schwarz Function and its Applications, The Carus Mathematical Monographs, 1974.
P. Duren, Theory of H p Spaces, Academic Press, New York, 1970.
P. Duren Univalent Functions, Springer, New York, 1983.
P. Ebenfelt, D. Khavinson and H. S. Shapiro, A free boundary problem related to single-layer potentials, Ann. Acad. Sci. Fenn. Math. 27 (2002) no.1, 21–46.
F. P. Gardiner, Teichmüller Theory and Quadratic Differentials, John Willey & Sons, 1987.
P. R. Garabedian, On the shape of electrified droplets, Comm. Pure Appl. Math. 18 (1965), 31–34.
P. R. Garabedian, E. McLeod, Jr. and M. Vitousek, Recent advances at Stanford in the application of conformal mapping to hydrodynamics, Amer. Math. Monthly 61 (1954) no.7 part II, 8–10.
B. Gustafsson, Application of half-order differentials on Riemann surfaces to quadrature identities for arc-length, J. Analyse Math. 49 (1987), 54–89.
B. Gustafsson and D. Khavinson, On approximation by harmonic vector fields, Houston J. Math. 20 (1994) no.1, 75–92.
J. A. Jenkins, Univalent Funcions and Conformal Mapping, Ergeb. Math. Grenzgeb [N.S.] 18, 1958.
D. Khavinson, Remarks concerning boundary properties of analytic functions of E p-classes, Indiana Math. J. 31(6) (1982), 779–781.
D. Khavinson, Symmetry and uniform approximation by analytic functions, Proc. Amer. Math. Soc. 101 (1987) no.3, 475–483.
D. Khavinson, An isoperimetric problem, in: Linear and Complex Analysis. Problem Book 3. Part II, Lecture Notes in Math., 1574, Springer, Berlin, 1994, 133–135, Springer-Verlag.
D. Khavinson, Smirnov classes of analytic functions in multiply connected domains, Appendix to English Transl. of S. Ya. Khavinson’s Foundations of the theory of extremal problems for bounded analytic functions and their various generalizations, Amer. Math. Soc. Transl., 129 (1986), 57–61
I. Laine, iNevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, 15 Walter de Gruyter & Co., Berlin, 1993.
E. B. McLeod, Jr., The explicit solution of a free boundary problem involving surface tension, J. Rational Mech. Anal. 4 (1955), 557–567.
—, An application of the Schiffer variation to the free boundary problems of hydrodynamics, Technical Report no. 15, Applied Mathematics and Statistics Laboratory, Stanford University, 1953.
Z. Nehari, Conformal Mapping, Dover Publications, Inc. New York, 1952.
L. E. Payne and P. W. Schaefer, Duality theorems in some overdetermined boundary value problems, Math. Methods Appl. Sci. 11 (1989), 805–819.
W. Reichel, Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains, Z. Anal. Anwendungen 15 (1996) no.3, 619–635.
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318.
H. S. Shapiro, The Schwarz Function and its Generalization to Higher Dimensions, John Willey & Sons, 1992.
B. Sirakov, Symmetry for exterior elliptic problems and two conjectures in potential theory, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001) no.2, 135–156.
G. C. Tumarkin and S. Ja. Khavinson, Classes of analytic functions on multiply connected domains, (in Russian) 1960, Issledovaniya po sovremennym problemam teorii funkci>ikompleksnogo peremennogo, 45–77, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow.
I. N. Vekua, Generalized Analytic Functions, Pergamon, London, 1962.
H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal. 43 (1971), 319–320.
S. N. B. Willms, G. M. L. Gladwell and D. Siegel, Symmetry theorems for some overdetermined boundary value problems on ring domains, Z. Angew. Math. Phys. 45 (1994) no.4, 556–579.
References
M. Miskis, J. M. Vanden-Broeck and J. Keller, Axisymmetric bubbles or drops in a uniform flow, J. Fluid. Mech. 108 (1980), 89–100.
P. N. Shankar, On the shape of a two-dimensional bubble in uniform motion, J. Fluid. Mech. 244 (1992), 187–200.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first author was supported by grant DMS-0139008 from the National Science Foundation; the research of the second author was supported by grant DMS-0412908 from the National Science Foundation.
Rights and permissions
About this article
Cite this article
Khavinson, D., Solynin, A.Y. & Vassilev, D. Overdetermined Boundary Value Problems, Quadrature Domains and Applications. Comput. Methods Funct. Theory 5, 19–48 (2005). https://doi.org/10.1007/BF03321084
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321084