Concerning Daniljuk's existence theorem for free boundary value problems with the Bernoulli condition

Author:
Andrew Acker

Journal:
Proc. Amer. Math. Soc. **80** (1980), 451-454

MSC:
Primary 49B21; Secondary 35R35, 76B99

DOI:
https://doi.org/10.1090/S0002-9939-1980-0581003-0

MathSciNet review:
581003

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that a variational method proposed by Ī. Ī. Daniljuk for proving the existence of free boundaries satisfying the Bernoulli condition is not valid under the condition given by Daniljuk.

**[1]**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**[2]**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**[3]**-,*Interior free boundary-value problems for the Laplace equation*, Arch. Rational Mech. Anal. (to appear).**[4]**A. Beurling,*On free-boundary problems for the Laplace equation*, Seminars on Analytic Functions, vol. 1, Inst. for Advanced Study, Princeton, N. J., 1957, pp. 248-263.**[5]**I. I. Daniljuk,*On a non-linear problem with a free boundary*, Dokl. Akad. Nauk SSSR**162**(1965), 979–982 (Russian). MR**0186825****[6]**I. I. Daniljuk,*An existence theorem in a certain nonlinear problem with free boundary*, Ukrain. Mat. Ž.**20**(1968), no. 1, 25–33 (Russian). MR**0224857****[7]**I. I. Daniljuk,*A certain class of nonlinear problems with free boundary*, Mathematical Physics, No. 7 (Russian), Naukova Dumka, Kiev, 1970, pp. 65–85 (Russian). MR**0273896****[8]**P. R. Garabedian,*Partial differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0162045****[9]**P. R. Garabedian, H. Lewy, and M. Schiffer,*Axially symmetric cavitational flow*, Ann. of Math. (2)**56**(1952), 560–602. MR**0053684**, https://doi.org/10.2307/1969661**[10]**D. Riabouchinsky,*Sur un problème de variation*, C. R. Acad. Sci. Paris Sér. A-B**185**(1927), 840-841.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0581003-0

Keywords:
Free boundary-value problem,
variational methods,
ideal fluid,
Bernoulli condition

Article copyright:
© Copyright 1980
American Mathematical Society