Rendiconti del Circolo Matematico di Palermo

, Volume 33, Issue 2, pp 255–264

# The extended Bitsadze-Lavrent’ev-Tricomi boundary value problem

• John M. Rassias
Article

## Abstract

F. G. Tricomi ([5], [6]) originated the theory of boundary of value problems for mixed type equations by establishing the first mixed type equation known asthe Tricomi equation $$y \cdot u_{xx} + u_{yy} = 0$$ which is hyperbolic fory<0, elliptic fory>0, and parabolic fory=0 and then observed that this equation could be applied in Aerodynamics and in general in Fluid Dynamics (transonic flows). See: M. Cribario [1], G. Fichera [2], and our doctoral dissertation [4]. Then M. A. Lavrent’ev and A. V. Bitsadze [3] established together a new mixed type boundary value problem for the equation$$\operatorname{sgn} (y) \cdot u_{xx} + u_{yy} = 0$$ where sgn (y)=1 fory>0, =−1 fory<0, fory=0, which involved thediscontinuous coefficient K=sgn (y) ofu xx while in the case of Tricomi equation the corresponding coefficientT=y wascontinuous. In this paper we establish another mixed type boundary value problem forthe extended Bitsadze-Lavrent’ev-Tricomi equation $$L u = \operatorname{sgn} (y) \cdot u_{xx} + \operatorname{sgn} (x) \cdot u_{yy} + r (x,y) \cdot u = f (x,y)$$ where both coefficientsK=sgn (y),M=sgn (x) ofu xx ,u yy , respectively are discontinous,r=r (x, y) is once continuously differentiable,f=f (x, y) continuous, and then we prove a uniqueness theorem for quasi-regular solutions.

## A.M.S. 1980 Mathematics Subject Classification

Primary: 35M05 Secondary: 76H05, 76N15, 76G99

## References

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