On the non-starlikeness of solutions to the starlike interior wake problem

Author:
Andrew Acker

Journal:
Proc. Amer. Math. Soc. **134** (2006), 749-753

MSC (2000):
Primary 35R35, 76B07

DOI:
https://doi.org/10.1090/S0002-9939-05-07991-8

Published electronically:
July 18, 2005

MathSciNet review:
2180893

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study examples of the starlike interior ``wake problem" for which no starlike solution exists relative to the natural star center of the problem. These examples show that the main result of D.E. Tepper in ``A mathematical model for a wake'' (Michigan Math. J. **31** (1984), 161-165) is not correct.

**[A1]**Andrew Acker,*Heat flow inequalities with applications to heat flow optimization problems*, SIAM J. Math. Anal.**8**(1977), no. 4, 604–618. MR**0473960**, https://doi.org/10.1137/0508048**[A2]**A. ACKER: Some free boundary optimization problems and their solutions. Numerische Behandlung von Differentialgleichungen mit besonderer Berucksichtigung freier Randwertaufgaben (J. Albrecht, L. Collatz, G. Hammerlin, editors), Birkhäuser Verlag, Basel 1978, pp. 9-22. (Proceedings of the conference by the same name held at the Mathematisches Forschungsinstitut Oberwolfach, Germany, May 1-7, 1977.) MR**0497989 (80m:35078)****[A3]**Andrew Acker,*A free boundary optimization problem involving weighted areas*, Z. Angew. Math. Phys.**29**(1978), no. 3, 395–408 (English, with German summary). MR**0482457**, https://doi.org/10.1007/BF01590761**[A4]**A. ACKER: On the qualitative theory of parametrized families of free boundaries. J. reine angew. Math. 193(1989), pp. 134-167. MR**0972364 (90a:35229)****[A5]**Andrew Acker,*Uniqueness and monotonicity of solutions for the interior Bernoulli free boundary problem in the convex, 𝑛-dimensional case*, Nonlinear Anal.**13**(1989), no. 12, 1409–1425. MR**1028238**, https://doi.org/10.1016/0362-546X(89)90102-8**[B1]**A. BEURLING: On free boundary problems for the Laplace equation. Seminars on Analytic Functions, Vol. I (1957), pp. 248-263. Institute for Advanced Study, Princeton, N.J.**[B2]**Arne Beurling,*The collected works of Arne Beurling. Vol. 1*, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1989. Complex analysis; Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. MR**1057613****[T1]**D.E. TEPPER: A free boundary problem in the annulus. J. Austral. Math. Soc. (Series A) 34(1983), pp. 177-181. MR**0687323 (84c:31001)****[T2]**D.E. TEPPER: A mathematical model for a wake. Mich. Math. J. 31(1984), pp. 161-165. MR**0752252 (86b:35205)****[T3]**D.E. TEPPER: A jet around an obstacle. In: Topics in Complex Analysis (Proceedings of the special session held at Fairfield, Conn., Oct. 28-29, 1983, D. Shaffer, editor). Contemporary Mathematics series, Vol. 38, American Mathematical Society, 1985, pp. 127-132. MR**0789455**

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Additional Information

**Andrew Acker**

Affiliation:
Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67208-0033

Email:
acker@math.wichita.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-07991-8

Keywords:
Ideal fluid,
free boundary,
geometric constraint,
cavitation,
starlikeness

Received by editor(s):
July 1, 2004

Received by editor(s) in revised form:
October 8, 2004

Published electronically:
July 18, 2005

Communicated by:
Richard A. Wentworth

Article copyright:
© Copyright 2005
American Mathematical Society