Electronic Journal of Differential Equations, Vol. 24 24, No. 93, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PROPERTIES OF SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS FOR LAVRENT EV-BITSADZE EQUATIONS AT CORNER POINTS MAKHMUD A. SADYBEKOV, NURGISSA A. YESSIRKEGENOV Abstract. We consider the Neumann-Tricomi problem for the Lavrent ev- Bitsadze equation for the case in which the elliptic part of the boundary is part of a circle. For the homogeneous equation, we introduce a new class of solutions that are not continuous at the corner points of the domain and construct nontrivial solutions in this class in closed form. For the nonhomogeneous equation, we introduce the notion of an n-regular solution and prove a criterion for the existence of such a solution.. Introduction Let R 2 be a finite domain bounded for y < by the characteristics AC : x + y = and BC : x y = of the Lavrent ev-bitsadze equation sgnyu xx + u yy = fx, y. and for y > by the circular arc σ δ = {x, y : x /2 2 +y+δ 2 = /4+δ 2, y > }. Neumann-Tricomi problem problem N-T. Find a solution of. with the boundary condition u AC =,.2 n σδ =,.3 where n = x /2 x + y + δ y. We assume that the classical transmission conditions ux, + = ux,, u y x, + = u y x,, < x <,.4 hold for the solution on the line y =, of type change of the equation. Along with problem N T, consider the adjoint problem. 2 Mathematics Subject Classification. 35M. Key words and phrases. Neumann-Tricomi problem; n-regular solution; Lavrent ev-bitsadze equation. c 24 Texas State University - San Marcos. Submitted July 9, 24. Published September 6, 24.
2 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 Problem N-T*. Find a solution of the equation with the boundary condition sgnyυ xx + υ yy = gx, y.5 υ BC =,.6 υ n σ δ =..7 Here we also assume that the transmission conditions υx, + = υx,, υ y x, + = υ y x,, < x <,.8 are satisfied. Bitsadze [, p. 34-37] proved the existence and uniqueness of regular solution of Neumann-Tricomi problem. The completeness of eigenfunctions of the Neumann- Tricomi problem for a degenerate equation of mixed type in the elliptic part of the domain was investigated by Moiseev and Mogimi [7]. Also, they showed that a system of functions consisting of sums of Legendre functions is complete. The existence and uniqueness of a strong solution of the Tricomi problem where instead of.3, it was given by condition u σ = for the Lavrent ev-bitsadze equation were studied in [3, 4, 5]. In [8] the spectral methods of solving boundary value problems for mixed-type differential equations of second order in a 3D domain were studied. Existence and uniqueness of a solution of the Lavrent ev Bitsadze problem was proved. In [] we proved a criterion for the strong solvability of the Neumann-Tricomi problem in L 2. It was shown that if the elliptic part of the domain coincides with the semi-circle, then the Neumann-Tricomi problem in the classical domain is not strongly solvable in L 2. In [] for the Tricomi problem it was studied properties of solutions at corner points. Also, it was given a criterion for the existence of n-regular solution. Relation between the uniqueness of solution of the problem and the order of smoothness or features of solutions is well-known and it is a particularly evident in problems for degenerate equations [9] and mixed-type equations []. In this paper, we introduce new classes of solutions of the Neumann-Tricomi problem depending on the behavior at the corner points and study their properties. 2. Main results We say that a function hx, y belongs to the class C α,β A,B if and only if x α x β hx, y C. As usual, = {y > } and 2 = {y < }. A solution of the problem is understood as a function in the class C 2 C 2 2 C C σ δ. We denote the angle at which the curve σ δ approaches the line of change of type satisfies γ δ = arccot2δ, < γ δ < π. 2. Theorem 2.. There are infinitely many solutions u k x, y C α k,α k A,B C σ δ to the homogeneous problem N-T f. These solutions are given by the relations u k x, y = { Re x+iy x 2 +y α k + Im x+iy 2 x 2 +y α k for y >, 2 x y α k 2.2 for y < ;
EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 3 where α k = π + 4k/4γ δ, k =,,... 2.3 In addition, u k x, y / L 2 for k, and u x, y L 2 only if γ δ > π/4. 2.4 Theorem 2.2. There are infinitely many solutions υ k x, y C α k, α k A,B C σ δ to the homogeneous problem N-T* g, where α k is given by relation 2.3. These solutions are given by the formulas υ k x, y = { Re x+iy α k + Im x+iy x 2 +y α k for y >, 2 x y α k 2.5 for y < ; x 2 +y 2 in addition, υ k x, y / L 2 for k, and υ x, y L 2 only under condition 2.4. Proof of Theorem 2... We denote x + iy wx, y = x 2 + y 2 αk, then ux, y = Re wx, y + Im wx, y, y >. For y > and f = equation. can be written as u xx + u yy =. By a direct calculation, we have x + iy αk 2 w xx = α k α k x 2 + y 2 x4 + 4 x 3 yi 6 x 2 y 2 4 xy 3 i + y 4 x 2 + y 2 4 + α k x + iy x 2 + y 2 αk 2 x 3 + 6 x 2 yi 6 xy 2 2y 3 i x 2 + y 2 3, x + iy αk 2 w yy = α k α k x 2 + y 2 x4 4 x 3 yi + 6 x 2 y 2 + 4 xy 3 i y 4 x 2 + y 2 4 + α k x + iy x 2 + y 2 α 2 x 3 6 x 2 yi + 6 xy 2 + 2y 3 i x 2 + y 2 3. Thus, w xx + w yy =, since x 2 + y 2 in, hence, Rew xx + w yy + Imw xx + w yy = u xx + u yy =. For y < and f = equation. can be written as u xx u yy =. By a direct calculation, from 2.2, for y < we have αk 2 u xx x, y = α k α k x y x y 4 αk + α k x y 2 x y 3,
4 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 αk 2 u yy x, y = α k α k x y x y 4 αk + α k x y 2 x y 3. Thus, u xx u yy =, since x + y in 2. The function in 2.2 satisfies equation. for both y > and y <. 2. By 2.2, for y <, ux, y = x y αk = x + y x y αk, we have x + y αk x+y= u AC = u x+y= = =, x y since α k >. Thus, function in 2.2 satisfies the boundary condition.2 in the hyperbolic part of the domain. The contour σ δ has the form 2yδ = x x 2 y 2. Thus, boundary condition.3 can be written as n x x 2 y 2 =2yδ =. By definition 2. for the number γ δ, we have n α k y α k σ δ = 2sin γ δ α k x2 + y 2 α cosα k γ δ sinα k γ δ. k By the definition of α k in 2.3, we obtain α k = π 4 + πk γ δ cosα k γ δ sinα k γ δ =. Thus, n σδ =. The function in 2.2 satisfies the boundary condition.3. 3. To check conditions.4, from the representation of 2.2, we obtain ux, = ux, + = x αk, u y x, = u y x, + = α k x αk x 2, and conditions.4 are satisfied. As a result, function in 2.2 is solution of the homogeneous Problem N-T. It is easy to see that function in 2.2 belongs to the class C α k,α k A,B C σ δ. Next, we prove the final statement of theorem 2.. u k 2 L = 2 u k x, y 2 dx dy <. 2.6 Note that u k x, y 2 dx dy = u k x, y 2 dx dy + 2 u k x, y 2 dx dy,
EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 5 u k x, y 2 x + y 2αk dx dy = dx dy 2 2 x + y 2αk < x + y dx dy. 2 Using the change of variables x + y = ξ, x y = η, we obtain u k x, y 2 dx dy < ξ 2α k dξdη = η dη ξ 2α k dξ 2 2 2 2 = η 2α k dη < 22α k for 2α k >, α k <. By using the change of variables x = r2 +r cos ϕ r sin ϕ +2r cos ϕ+r, y = 2 +2r cos ϕ+r in 2, we obtain u k x, y 2 r 2αk + sin 2α k ϕr dx dy = + 2r cos ϕ + r 2 dr dϕ 2 = γδ + sin 2α k ϕdϕ r 2α k+ dr + 2r cos ϕ + r 2 2 < for 2α k + + 4 < α k <. Thus, ratio 2.6 is satisfied for α k <. Taking into account the definition of α k in 2.3, it is easy to see that ratio 2.6 is satisfied only for k =, and π 4γ δ < γ δ > π 4. Theorem 2.2 can be proved in a similar way; so we omit its proof. Let us proceed to the analysis of the nonhomogeneous problem N-T and N-T*. An n-regular solution of Problem N-T N-T* is defined as a solution, ux, y C 2 C 2 2 C C σ δ C n, A,B υx, y C 2 C 2 2 C C σ δ C, n A,B. The following theorems hold for the nonhomogeneous Problems N-T and N-T*. Theorem 2.3. The solution of Problem N-T is n-regular for any right-hand side fx, y C if and only if γ δ < π/4n, n =, 2,... 2.7 The solution is n-regular for arbitrary approach angles γ δ only if the right-hand side of. satisfies the conditions υ k x, yfx, y dx dy =, k =,..., j, 2.8 where υ k are the functions given by 2.5, j = [nγ δ /π /4], and [z] is the integer part of z. In this case, the number of conditions 2.8 depends on the angle γ δ, and their maximum number is equal to n as γ δ π.
6 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 Theorem 2.4. Condition 2.7 is necessary and sufficient for the n-regularity of the solution of Problem N-T* for any right-hand side gx, y C; for arbitrary approach angles γ δ, the right-hand side of.5 satisfies the relations u k x, ygx, y dx dy =, k =,..., j, 2.9 where the u k are the functions given by 2.2 and j = [nγ δ /π /4]. In this case, the number of conditions 2.9 depends on the angle γ δ, and their maximum number is equal to n as γ δ π. Remark 2.5. Conditions 2.8 and 2.9 with k are not orthogonality conditions in L 2, and for k they are orthogonality conditions only if inequality 2.4 holds. This immediately follows from theorems 2. and 2.2. Proof of Theorem 2.3. Set ux, y = τx and u y x, = νx. In the hyperbolic part of the domain 2, we consider the Cauchy-Goursat problem u xx + u yy = fx, y, u AC =, u y x, = νx. The solution of this problem has the form [2, p.2]: x+y x+y ux, y = νtdt x y dξ f ξ + η, ξ η dη. 2 ξ 2 2 Then we obtain the main relation x τx = νtdt x x dξ f ξ + η, ξ η dη, < x <. 2 ξ 2 2 It is convenient to represent it in the form x τ = x + x νx θ/ + x θ + x θ 2 dθ + F x + x, < x <, 2. where F x = x x dξ f ξ + η, ξ η dη. 2 ξ 2 2 We apply the Mellin transform F s = x s fxdx to both sides of relation 2.. By using the formula [2, p. 269] x s dx x for ux = νx/+x +x 2 and uxθυθdθ = υx = from 2., we obtain the relation x s uxdx { for x <, for x >, x s υxdx, τs = s νs + F s. 2. Here τs = x s τ x dx, νs = x s νx/ + x x + x + x 2 dx, 2.2 F s = x s x F dx. 2.3 + x
EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 7 In the elliptic part, we consider the problem u xx + u yy = fx, y, By making the change of variables x = n σδ =, ux, = τx. r2 + r cos ϕ + 2r cos ϕ + r 2, y = r sin ϕ + 2r cos ϕ + r 2, 2.4 by using the Mellin transform, and by solving the resulting problem, we obtain where fs, ϕ = νs = tansγ δ sτs γδ u 2 s, tfs, tdt, 2.5 r s r 2 + 2r cos ϕ + r 2 2 f r 2 + r cos ϕ + 2r cos ϕ + r 2, r sin ϕ + 2r cosϕ + r 2 dr, 2.6 u 2 s, ϕ = cos sϕ + sin sϕ sin sγ δ cos sγ δ, 2.7 and the functions τs and νs are defined in 2.2. Now from relations 2. and 2.5, we obtain [ γδ ] νs = tansγ δ sf s u 2 s, tfs, tdt [ + tansγ δ ], 2.8 τs = [ sf s + γδ ] u 2 s, tfs, tdt [s + tansγ δ ]. 2.9 First, let us analyze definitions 2.2 of the functions τs and νs and their relationship with the original functions τx and νx. By making the obvious change of variables t = x/ + x, we reduce relation 2.2 to the form τs = t s t s τtdt, νs = t s t s νtdt. Hence, it follows that if the function τs is continuous on the interval,, then the function τt is continuous at the point t = and has a zero of order there. As a result, by taking into account the definitions of the functions τx and νx, for the n-regularity of the solution, the right-hand sides in relations 2.8 and 2.9 should be continuous for n < s <, whence we obtain γ δ < π/4n. Consequently, condition 2.7 is necessary and sufficient for the n-regularity of the solution of the Neumann-Tricomi problem for any right-hand side fx, y C. The proof of first part of Theorem 2.3 is completed. Now let us proceed to the proof of properties of solutions for arbitrary approach angles γ δ. Suppose that condition 2.7 fails. It follows that, for s = α k = π + 4k/4γ δ n,, for k =,..., j, and for j, which is defined in the statement of the theorem, the denominator in relations 2.8 and 2.9 is zero. Therefore, for the n-regularity, it is necessary and sufficient that the numerator is zero at these points as well, γδ sf s + u 2 s, tfs, tdt =. 2.2 s= αk
8 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 In this equation, we take into account relation 2.3 and the following property of the Mellin transform [6, p.567]: if then gs = sgs = x s fx dx, x s xf x dx. Now, by setting s = α k, by returning to the variables x and y according to formulas 2.4, and by taking into account relations 2.3, 2.5, 2.6, and 2.7, we find that condition 2.2 can be represented in the form υ k x, yfx, ydxdy + υ k x, yfx, ydxdy + = υ k x, yfx, ydxdy =. Here k =,..., j ; moreover, the number of such k by the definition of j cannot exceed n. Theorem 2.4 can be proved in a similar way, we omit its proof. Conclusion. In this article, it has been shown that the number of solutions of the homogeneous Neumann-Tricomi problem admitting a feature at the corner points of the domain, depends on the order of the singularity and depends on the order of the singularity and on the values of approach angles of an elliptic part of boundary of the domain to the line change of type. We have shown that for what angles of approach a singular solution of homogeneous Neumann-Tricomi problem belongs to the space L 2. In case of Neumann-Tricomi problem, unlike Tricomi problem in space L 2, only the value of angle at point A solves everything and the angle of approach at point B does not react []. Also, we have obtained conditions of existence of n-regular solutions for the nonhomogeneous Neumann-Tricomi problem. These conditions have been formulated in terms of orthogonality of the function in the right hand side of the equation to the corresponding singular solutions of its adjoint homogeneous problem. Acknowledgements. The authors are grateful to Professor T. Sh. Kalmenov for his support and attention to this work. This work has been supported by Committee of Science of Ministry of Education and Science of the Republic of Kazakhstan grant number 743. References [] A. V. Bitsadze; On the problem of equations of mixed type. Trudy Mat. Inst. Steklov., 4 953, pp. 3-59. [2] A. Erd elyi, W. Magnus, F. Oberherhettinger, F. G. Tricomi; Tablititsy integral nykh preobrazovaninii. T.. Preobrazovaniya Fur e, Laplasa, Mellina Tables of Integral Transforms. Vol.. Fourier, Laplace, Mellin Transforms. Moscow: Nauka, 969. [3] T. Sh. Kalmenov, A. B. Bazarbekov; A Criterion for the Strong Solvability of a Tricomi Problem for the Lavrent ev-bitsadze Equation. Dokl. Akad. Nauk SSSR, 26 2 98, pp. 265-268. [4] T. Sh. Kalmenov, A. B. Bazarbekov; A Criterion for the Strong Solvability of a Tricomi Problem for the Lavrent ev-bitsadze Equation in L p. Differential Equations, 8 2 982, pp. 268-28.
EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 9 [5] T. Sh. Kalmenov, M. A. Sadybekov, N.E. Erzhanov; A Criterion for the Strong Solvability of the Tricomi Problem for the Lavrent ev-bitsadze Equation. The General Case. Differential Equations, 29 5 993, pp. 87-875. [6] M. A. Lavrent ev, B. V. Shabat; Metody therii funktsii kompleksnogo premennogo Methods of the Theory of Functions of a Complex Variable. Moscow, 967. [7] E. I. Moiseev, M. Mogimi; On the Completeness of Eigenfunctions of the Neumann-Tricomi Problem for a Degenerate Equation of Mixed Type. Differential Equations, 4 2 25, pp. 789-79. [8] E. I. Moiseev, P. V. Nefedov; Tricomi problem for the Lavrent ev Bitsadze equation in a 3D domain. Integral Transforms and Special Functions, 23 22, pp. 76-768. [9] L. S. Pulkina; Certain nonlocal problem for a degenerate hyperbolic equation. Mathematical Notes, 5 3 992, pp. 286-29. [] A. V. Rogovoy; Properties of solutions of the Tricomi problem for the Lavrent ev-bitsadze equation at corner points. Differential Equations, 49 2 23, pp. 65-654. [] M. A. Sadybekov, N. A. Yessirkegenov; Strong solvability criterion of Neumann-Tricomi problem in L 2 for the Lavrent ev-bitsadze equation. Mathematical Journal, 47 23, pp. 63-72. [2] M. M. Smirnov; Uravneniya smeshannogo tipa Equations of Mixed Type. Moscow: Vyssh. Shkola 985. Makhmud A. Sadybekov Institute of Mathematics and Mathematical Modeling, 25 Pushkin str., 5 Almaty, Kazakhstan E-mail address: makhmud-s@mail.ru Nurgissa A. Yessirkegenov Institute of Mathematics and Mathematical Modeling, 25 Pushkin str., 5 Almaty, Kazakhstan. Al-Farabi Kazakh national university, 7 ave. Al-Farabi, 54 Almaty, Kazakhstan E-mail address: nurgisa@hotmail.com