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On starshaped rearrangement and applications


Author: Bernhard Kawohl
Journal: Trans. Amer. Math. Soc. 296 (1986), 377-386
MSC: Primary 35R35; Secondary 26D20
DOI: https://doi.org/10.1090/S0002-9947-1986-0837818-4
MathSciNet review: 837818
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Abstract: A radial symmetrization technique is investigated and new properties are proven. The method transforms functions $ u$ into new functions $ {u^\ast}$ with starshaped level sets and is therefore called starshaped rearrangement. This rearrangement is in general not equimeasurable, a circumstance with some surprising consequences. The method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems. In particular, the Lipschitz continuity of free boundaries can now be easily obtained in a new fashion.


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  • [1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144. MR 618549 (83a:49011)
  • [2] C. Bandle and M. Marcus, Radial averaging transformation with various metrics, Pacific J. Math. 46 (1973), 337-348. MR 0352502 (50:4989)
  • [3] -, Radial averaging transformations and generalized capacities, Math. Z. 145 (1975), 11-17. MR 0393522 (52:14331)
  • [4] J. I. Diaz and M. A. Herrero, Estimate on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 347-359. MR 635761 (83i:35019)
  • [5] H. Federer, Curvature measure, Trans. Amer. Math. Soc. 93 (1959), 418-491. MR 0110078 (22:961)
  • [6] L. E. Fraenkel and M. S. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13-51. MR 0422916 (54:10901)
  • [7] L. S. Frank and W. D. Wendt, On an elliptic operator with discontinuous nonlinearity, J. Differential Equations 54 (1984), 1-18. MR 756543 (86b:35061)
  • [8] A. Friedman, Variational principles and free boundary problems, Wiley, New York, 1982. MR 679313 (84e:35153)
  • [9] A. Friedman and D. Phillips, The free boundary of a semilinear elliptic equation, Trans. Amer. Math. Soc. 282 (1984), 153-182. MR 728708 (85d:35042)
  • [10] H. Grabmüller, A note on equimeasurable starshaped rearrangement or: "How long is John Miller's nose?", Report 096, Erlangen, 1983.
  • [11] B. Kawohl, Starshapedness of level sets for the obstacle problem and for the capacitory potential problem, Proc. Amer. Math. Soc. 89 (1983), 637-640. MR 718988 (86c:35063)
  • [12] -, A geometric property of level sets of solutions to semilinear elliptic Dirichlet problems, Appl. Anal. 16 (1983), 229-234. MR 712734 (85f:35083)
  • [13] -, Geometrical properties of level sets of solutions to elliptic problems, Proc. Sympos. Pure Math., vol. 45, part 2, Amer. Math. Soc., Providence, R.I., 1986, pp. 25-36. MR 843592 (88i:35044)
  • [14] -, Starshaped rearrangement and applications, LCDS Report 83-20, Brown Univ., Providence, R.I., 1983.
  • [15] M. Marcus, Transformations of domains in the plane and applications in the theory of functions. Pacific J. Math. 14 (1964), 613-626. MR 0165093 (29:2382)
  • [16] -, Radial averaging of domains, estimates for Dirichlet integrals and applications, J. Analyse Math. 27 (1974), 47-93. MR 0477029 (57:16573)
  • [17] D. S. Mitrinović, Analytic inequalities, Springer, Berlin, 1970. MR 0274686 (43:448)
  • [18] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967.
  • [19] L. E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 721-729. MR 0333487 (48:11812)
  • [20] L. E. Payne and A. Weinstein, Capacity, virtual mass, and generalized symmetrization, Pacific J. Math. 2 (1952), 633-641. MR 0050738 (14:375b)
  • [21] D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J. 32 (1983), 1-17. MR 684751 (84e:49012)
  • [22] J. A. Pfaltzgraff, Radial symmetrization and capacities in space, Duke Math. J. 34 (1967), 747-756. MR 0222324 (36:5376)
  • [23] G Polya and G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. Math. Stud., no. 27, Princeton Univ. Press, Princeton, N.J., 1951. MR 0043486 (13:270d)
  • [24] R. Sperb, Extension of two theorems of Payne to some nonlinear Dirichlet problem, Z. Angew. Math. Phys. 26 (1975), 721-726. MR 0393835 (52:14643)
  • [25] J. Spruck, Uniqueness in a diffusion model of population biology, Comm. Partial Differential Equations 8 (1983), 1605-1620. MR 729195 (85h:35086)
  • [26] G. Szegö, On a certain kind of symmetrization and its applications, Ann. Mat. Pura Appl. (4) 40 (1955), 113-119. MR 0077664 (17:1074b)
  • [27] D. F. Tepper, Free boundary problem, SIAM J. Math. Anal. 5 (1974), 841-846. MR 0361132 (50:13578)
  • [28] -, On a free boundary problem, the starlike case, SIAM J. Math. Anal. 6 (1975), 503-505. MR 0367237 (51:3479)

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DOI: https://doi.org/10.1090/S0002-9947-1986-0837818-4
Article copyright: © Copyright 1986 American Mathematical Society

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