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A uniqueness theorem for the Chaplygin-Frankl problem

Published online by Cambridge University Press:  17 April 2009

John M.S. Rassias
John M.S. Rassias
Affiliation:
II Dervenakion Str., Daphne, Athens, Greece.
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Abstract

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In a paper dealing with trans-sonic jet flows Frankl (Bull. Acad. Sci. URSS Sér. Math. [Izv. Akad. Nauk SSSE] 9 (1945), 121–143) considered the following problem (T) by applying the condition

where k = k(y) is a monotone increasing function with a continuous second derivative, k(0) = 0, F(0) > 0, k′(y) ≠ 0 for y < 0. Consider an equation of the form

which is elliptic for y > 0, hyperbolic for y < 0, and parabolic for y = 0. Consider equation (2) in a bounded simply connected region D ⊂ R2 which is bounded by the following three curves: a piecewise smooth curve Γ0 lying in the half-plane y > 0 which intersects the line y = 0 at the points A(0, 0) and B(l, 0); for y < 0 by a smooth curve Γ2 through B which meets the characteristic of (2) issuing from A(0, 0) at the point P; and the curve Γ1 which consists of the portion PA of the characteristic through A. The problem (T) (or problem of Tricomi-Frankl) consists of finding a solution u = u(x, y) ∈ C2(D) assuming prescribed values on Γ0 ∪ Γ2. In the present paper we generalize Frankl's uniqueness theorem; our uniqueness theorem includes cases where F(y) may be negative.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 2 , April 1979 , pp. 217 - 226
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Франль, Ф.И. [Frankl, F.], “О задачах C.A. Чаплыгина для смешанных до- и сверхзвуновых течений” [On the problems of Chaplygin for mixed sub- and supersonic flows], Bull. Acad. Sci. URSS Sér. Math. [Izv. Akad. Nauk SSSR] 9 (1945), 121–143.Google Scholar
[2]Kapilevich, M.B., “Equations of mixed elliptic and hyperbolic type”, Linear equations of mathematical physics, 215–243 (Holt, Rinehart and Winston, New York, Chicago, San Francisco, Toronto, London, 1967).Google Scholar
[3]Protter, M.H., “Uniqueness theorems for the Tricomi problem”, J. Rational Mech. Anal. 2 (1953), 107–114.Google Scholar
[4]Rassias, John Michael, “Mixed type partial differential equations in Rn” (PhD dissertation, University of California, Berkeley, 1977).Google Scholar