Uniqueness Theorems for the Frankl-Morawetz Problem in R2 for a Class of Nonlinear Equations of Mixed Type

We consider the question of uniqueness for the solutions of the boundary value problem (1) where sgn K(y) = sgn y and Γ₀ and Γ₁ are parts of the boundary of a bounded simply connected region G in R². G is bounded for y > 0 by a piecewise smooth curve Γ₀ which intersects the line y = 0 at A (– 1, 0) and B(0, 0). For y < 0 G is bounded by a piecewise smooth curve Γ₁ through A, which meets the characteristic of (1) issued from B at point C, and by the curve Γ₂ which consists of the portion CB of the characteristic through B. Using energy-integral considerations, we give sufficient conditions for the uniqueness of solutions to boundary value problem (1).