# Gellerstedt problem

A boundary value problem for a Chaplygin-type equation

in which the function increases, and for . The function which is to be found is specified on the boundary. This boundary consists of a sufficiently-smooth contour and pieces of characteristics. This equation is elliptic in the half-plane , parabolic on the line , and hyperbolic for . The half-plane of hyperbolicity is covered by two families of characteristics, which satisfy the equations and .

Figure: g043640a

The characteristics of one of these families merge with the characteristics of the other on the line .

Let be a simply-connected domain with as boundary a sufficiently-smooth contour if or pieces , , , and if , and being the characteristics of one family, and and of the other (see Fig.). The theorem on the existence and the uniqueness of solutions of the following boundary value problems is valid in : the function is given on ; the function is given on .

These problems were first studied (for , ) by S. Gellerstedt [1] by methods developed by F. Tricomi [2] for the Tricomi problem, and represent a generalization of that problem. Gellerstedt's problem has important applications in gas dynamics with velocities around the velocity of sound. These and related problems were studied for certain multiply-connected domains and for linear equations containing lower-order terms [3].

#### References

 [1] S. Gellerstedt, "Quelques problèmes mixtes pour l'équation " Ark. Mat. Astr. Fysik , 26A : 3 (1937) pp. 1–32 [2] F.G. Tricomi, "Integral equations" , Interscience (1957) [3] M.M. Smirnov, "Equations of mixed type" , Amer. Math. Soc. (1978) (Translated from Russian)