eu ON UNIQUENESS GENERALIZED PROBLEM OF TRICOMI FOR THE CHAPLYGIN EQUATION

Abstract: In this paper we consider the generalized Tricomi problem in a mixed domain for the Chaplygin equation. Frankl was first which showed that the problem of the expiry of a supersonic jet from a vessel with plane walls in the hodograph plane is reduced to the Tricomi problem for Chaplygin equation. By method of auxiliary functions we received a new theorem of uniqueness of the solution of this problem without any restrictions, except the smoothness on the elliptical part of the border domain.


Consider the equation
Lz = K(y)z xx + z yy = 0, (1) in an open domain D, where yK(y) > 0 for y = 0.The domain D is bounded by curves: a piecewise smooth curve Γ in the half-plane y > 0, which intersects the line y = 0 at the points A(0, 0) and B(l, 0), l > 0; in y < 0, D is bounded by two curves: monotonic curve γ 1 : y = α(x), (α ′ (x) < 0) issuing from A and meeting at the point C with charasteristic y c , 0), y c -the ordinate of point C and let D + be subdomain of D with y > 0 and D − be subdomain of D with y < 0.
In this paper using a variation of the energy-integral method (abc method) we obtain sufficient conditions for the uniqueness of solution of generalized problem of Tricomi for the Chaplygin equation.It arises in the study of transonic flow, and the proof of uniqueness in this case leads to a proof that continuous transonic flows past smooth profilies do not exist in general [1].
The generalized problem of Tricomi.Find function z(x, y) satisfying the following conditions: where ϕ and ψ are given functions, L -length of curve Γ. iii) the boundary integrals which arise exist in the sense that: the limits taken over corresponding interior curves exist as these interior curves approach the boundary.
The question uniqueness of solution of generalized problem of Tricomi for equation of mixed type has been dealt with in the literature by many authors.For an extensive bibliography we refer the reader to [1], [2], [3].

Theorem of Uniqueness
We introduce Francl's function The following statement is a more general result than Theorem 6, given in [2].
Proof.Consider the area integral I over domain D where a(x, y), b(x, y), c(x, y) are given functions.By (1), the integral I vanishes.We shall show that over D integral I can be made non-positive by proper choice of functions a(x, y), b(x, y) and c(x, y).Applying Green's formula to the integral (2), similar to the work [3], we get: Choose b = c ≡ 0 in D + .From z(x, y) = 0 on Γ ∪ γ 1 and the fact that dx = √ −Kdy (on γ 2 ) and z x dx + z y dy = 0 γ 1 we get We must choose functions a(x, y), b(x, y) and c(x, y) so that all the integrals I 1 , I 2 , ..., I 6 or at least their partial combinations were non-positive.If this occurs then z(x, y) ≡ 0 follows immediately.
Obviously I 2 = 0.An integration by parts I 3 we get The integral I 6 is non-positive if the following two conditions hold in D − : 2a + b x − c y ≥ 0.

Definition 1 .
We call a function regular solution of (1) if the following hold i) z(x, y) ∈ C(D) ∩ C 1 (D) ∩ C 2 (D − ∪ D + ); ii) we can to applicate Greens theorem to the integrals D zLzdxdy D zLz x dxdy D z y Lzdxdy;