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 Free Access

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]**Andrew Acker,*Some free boundary optimization problems and their solutions*, Numerische Behandlung von Differentialgleichungen mit besonderer Berücksichtigung freier Randwertaufgaben (Tagung Math. Forschungsinst., Oberwolfach, 1977) Internat. Ser. Numer. Math., vol. 39, Birkhäuser, Basel-Boston, Mass., 1978, pp. 9–22. MR**497989****[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

Andrew Acker,*Another free boundary optimization problem involving weighted areas*, Z. Angew. Math. Phys.**29**(1978), no. 3, 409–413 (English, with German summary). MR**0482458**, https://doi.org/10.1007/BF01590762**[A4]**Andrew Acker,*On the qualitative theory of parametrized families of free boundaries*, J. Reine Angew. Math.**393**(1989), 134–167. MR**972364****[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**

Arne Beurling,*The collected works of Arne Beurling. Vol. 2*, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1989. Harmonic analysis; Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. MR**1057614****[T1]**David E. Tepper,*A free boundary problem in an annulus*, J. Austral. Math. Soc. Ser. A**34**(1983), no. 2, 177–181. MR**687323****[T2]**David E. Tepper,*A mathematical model for a wake*, Michigan Math. J.**31**(1984), no. 2, 161–165. MR**752252**, https://doi.org/10.1307/mmj/1029003020**[T3]**David E. Tepper,*A jet around an obstacle*, Topics in complex analysis (Fairfield, Conn., 1983) Contemp. Math., vol. 38, Amer. Math. Soc., Providence, RI, 1985, pp. 127–132. MR**789455**, https://doi.org/10.1090/conm/038/16

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35R35,
76B07

Retrieve articles in all journals with MSC (2000): 35R35, 76B07

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