Oblique Derivative Problem for Generalized Lavrent′ev-Bitsadze Equations

In this article, we first give the representation of solutions for the oblique derivative boundary value problem of generalized Lavrent′ev-Bitsadze equations including the Lavrent′ev-Bitsadze equation. Next we verify the uniqueness of solutions of the above problem. Finally we prove the solvability of oblique derivative problems for quasilinear mixed (generalized Lavrent′ev-Bitsadze) equations of second order, at the same time the estimates of solutions of the above problem is also obtained. The above problem is an open problem proposed by J. M. Rassias.

Let D be a simply connected bounded domain in the complex plane C with the boundary ∂D = Γ ∪ L, where 3R} is a hyperbolic curse, and denote by D + = D ∩ {ŷ > 0}, D − = D ∩ {ŷ < 0} the elliptic domain and hyperbolic domain respectively, z 0 = 2iR, and z 1 = −2 j √ 2R is the intersection point of L 1 , L 2 , where i is the imaginary unit and j is the hyperbolic unit with the condition j 2 = 1.Here the common boundary of elliptic domain and hyperbolic domain is a hyperbolic curse, which has not been discussed in our previous papers, and in Wen (2013), the common boundary of elliptic domain and hyperbolic domain is a circle.We consider the second order quasilinear equation of mixed type u xx + sgnŷ u yy = au x + bu y + cu + d in D, (1.1) where ŷ = y − 3R + √ R 2 + x 2 , a, b, c, d are functions of z(∈ D), u, u x , u y (∈ R), its complex form is the following complex equation of second order where A j = A j (z, u, u z ), j = 1, 2, 3, and In which we use the complex number z = x + iy in D + and the hyperbolic number z = x + jy in D − with the hyperbolic unit j.In this article, the notations are as the same in References (Wen, 1986(Wen, , 1992(Wen, , 2002(Wen, , 2008(Wen, , 2010(Wen, , 2013;;Wen, Chen, & Xu, 2008;Huang, Qiao, & Wen, 2005).
Problem P The oblique derivative boundary value problem for (1.2) is to find a continuously differentiable solution u(z) of (1.2) in D * = D\{z 1 , z 2 }, which is continuous in D and satisfies the boundary conditions 1 2 where ν is a given vector at every point on is called the index of Problem P and Problem P 0 , where in which [a] is the largest integer not exceeding the real number a, t 1 = z * , t 2 = z * on L 0 , here we only discuss the case of K = 0 on ∂D + , and the solution of Problem P is unique.
Besides, if the index K = 1/2, we can add a point condition where z 2 is an inner point of Γ, b 3 is a real constant with the condition |b 3 | ≤ k 2 , and the boundary value problem for (1.2) will be called Problem Q.
Setting that it is clear that Problem P for (1.2) is equivalent to the Riemann-Hilbert boundary value problem (Problem A) for the first order complex equation of mixed type with the boundary conditions (1.12) and the relation From the formula (2.10), Chapter II, Wen (2002), we see that the above integral is independent of integral path in D and u(z) is continuously differentiable in D * = D\{z * , z * }.
Obviously the Lavrent ev-Bitsadze equation is a special case of generalized Lavrent ev-Bitsadze Equation (1.1), the relation of u(z) and w(z) is as stated in (1.10) and (1.13).

Solvability of Oblique Derivative Problem for Lavrent ev-Bitsadze Equation
We first prove the existence and representation of solutions for Problem A of the equation with the boundary conditions on L 1 is a known real function, r 0 (z) on L 0 is an undetermined real constant, b 2 is a real constant, and λ(z), r(z), b 2 satisfy the conditions in which α (0 < α < 1), k 0 , k 2 (≥ k 0 ) are positive constants; and From the boundary condition (1.15) of Tricomi problem in Section 1, we can find the directive derivation for (1.15) according to the arc length parameter s on Γ ∪ L 1 , and get the boundary conditions of oblique derivative problem (Problem P) as follows Later on we shall use the condition on L 1 .
For this, we shall find the solution of the last system of (2.1) in D − with the boundary conditions (2.6) In fact the solution of Problem A for (2.1) in D − can be expressed as ]/2, and in which f (t), g(t) are two arbitrary real continuous functions on , thus the formulas in (2.6) can be rewritten as and can find U(z 1 ) = (a(z 1 )r(z 1 ) (2.9) Thus we can derive Due to the above formula (2.5), we can obtain the oblique derivative condition of Tricomi problem as follows (2.11) x 2 into the formula (2.10), we obtain the boundary condition (2.12) where In addition, from the above condition and the first boundary condition in (2.2), noting that the index K = 0, there exists a unique solution w(z) = U + iV of discontinuous Riemann-Hilbert problem with the boundary conditions (2.12) and the first condition in (2.2) for the Equation (1.14) in D + (see Theorem 6.6, Chapter V, Wen, 1992), and then the solution of Problem A for (1.14) is obtained as follows (2.13) where Hence we have the following theorem.
Theorem 2.1 Problem A for (2.1) in D has a unique solution in the form (2.13), which satisfies the estimates ) ε ) are positive constants.Let the solution w(z) of Problem A for (1.14) be substituted in (1.13).Then the solution u(z) of Problem P for Lavrent ev-Bitsadze Equation (1.14) is obtained, which can represented by the formula (1.13).We mention that the method in this section is completely solved the solvability of oblique derivative problem for Lavrent'ev-Bitsadze equation, which is differed with our previous researches including (Wen, 2013), and the reasoning is stringent and very intersenting.
In brief, the proof of the solvability for Problem P of (1.14) can be divided into four steps: (1) From the second and third conditions in (2.2) for the Equation (1.14) in D − , the boundary condition (2) From the first boundary condition in (2.2) and the above condition (2.15), the continuous solution w(z) of Problem A in D + \{z * , z * } is obtained, at the same time we determine the boundary condition (2.16) (3) From the boundary conditions in (2.2) and (2.16), we can find the solution w(z) of Problem A in D − as stated in (2.13).
(4) To substitute the solution w(z) of Problem A for the Equation (1.14) into the formula (1.13), thus the solution u(z) of the oblique derivative boundary value problem (Problem P) for the Lavrent ev-Bitsadze Equation (1.14) is gotten.

Unique Solvability for Problem P for Generalized Lavrent ev-Bitsadze Equations
First of all we give the representation theorem of Problem P for the Equation (1.2).