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Open problems connected with level sets of harmonic functions

Collection of Problems

Part of the Lecture Notes in Mathematics book series (LNM,volume 1344)

Keywords

  • Harmonic Function
  • Free Boundary
  • Isoperimetric Inequality
  • Free Boundary Problem
  • Inverse Function Theorem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. ACKER, A.: On the geometric form of free boundaries satisfying a Bernoulli boundary condition. I and II. Math.Methods Appl. Sci. 6 (1984) 449–456 and 8 (1986) 378–404

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. ALT, H. W. & L. A. CAFFARELLI: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981) 105–144

    MathSciNet  MATH  Google Scholar 

  3. GERGEN, J. J.: Note on the Green function of a starshaped threedimensional region. Amer. J. Math. 53 (1931) 746–752

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. HAMILTON, R. S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7 (1982) 65–222

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. KAWOHL, B.: Rearrangements and convexity of level sets in PDE. Springer Lecture Notes in Math. 1150 (1985)

    Google Scholar 

  6. KAWOHL, B.: Some qualitative properties of nonlinear partial differential equations. Proceedings of the MSRI microprogram on nonlinear diffusion equations and their equilibrium states, Berkeley, 1986, Eds. W. M. Ni, J. Serrin & L. Peletier, to appear

    Google Scholar 

  7. LEWIS, J.: Capacitary functions in convex rings. Arch. Ration. Mech. Anal. 66 (1977) 201–224

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. NIKLIBORC, W.: Úber die Niveaukurven logarithmischer Flächenpotentiale. Math. Zeitschr. 36 (1933) 641–646

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. PFALTZGRAFF, J. A.: Radial symmetrization and capacities in space. Duke Math. J. 34 (1967) 747–756

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. POLYA, G.: Liegt die Stelle der größten Beanspruchung an der Oberfläche? Zeitschr. Angew. Math. Mech. 10 (1930) 353–360

    CrossRef  MATH  Google Scholar 

  11. POLYA, G. & G. SZEGÖ: Isoperimetric inequalities in mathematical physics. Annals of Math. Studies 27 (1951) Princeton Univ. Press

    Google Scholar 

  12. SPERB, R.: Maximum principles and their applications. Acad. Press, New York, 1981

    MATH  Google Scholar 

  13. STODDART, A. W. J.: The shape of level surfaces of harmonic functions in three dimensions. Michigan Math. J. 11 (1964) 225–229

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. TEPPER, D. F.: Free boundary problem. SIAM J. Math. Anal. 5 (1974) 841–846

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. WARSCHAWSKI, S. E.: On the Green function of a starshaped threedimensional region. Amer. Math. Monthly 57 (1950) 471–473

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. WEATHERBURN, C. E.: On families of surfaces. Math. Annalen 99 (1928) 473–478

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1988 Springer-Verlag

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Kawohl, B. (1988). Open problems connected with level sets of harmonic functions. In: Král, J., Lukeš, J., Netuka, I., Veselý, J. (eds) Potential Theory Surveys and Problems. Lecture Notes in Mathematics, vol 1344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103356

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  • DOI: https://doi.org/10.1007/BFb0103356

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50210-4

  • Online ISBN: 978-3-540-45952-1

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