A boundary-value problem for the Tricomi equation in a special domain
The equation of mixed type NEWLINE\[NEWLINE\text{sgn} y\cdot |y|^m u_{xx}+ u_{yy}=0, \quad m>0 NEWLINE\]NEWLINE is considered in the domain bounded for \(y\geq 0\) by the normal curve NEWLINE\[NEWLINE \Gamma=\{(x,y): (x-1/2)^2+4(m+2)^{-2}y^{m+2}=1/4, 0\leq x\leq 1\} NEWLINE\]NEWLINE and for \(y<0\) by the characteristics NEWLINE\[NEWLINE AC_1=\{(x,y): \xi=x-2(m+2)^{-1}(-y)^{(m+2)/2}=0, 0< x\leq 1/4\}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE BC_2=\{(x,y): \eta=x+2(m+2)^{-1}(-y)^{(m+2)/2}=1, 3/4\leq x< 1\} NEWLINE\]NEWLINE of the equation and the segment \(1/4<x<3/4\) of the line \(y=-((m+2)/8)^{2/(m+2)}\). The uniqueness and the existence of the solution of the problem with the boundary conditions NEWLINE\[NEWLINE u|_{\Gamma}=\varphi(s),\quad u|_{AC_1}=\psi(\eta), \quad \partial u/\partial y|_{C_1C_2}=\chi(x) NEWLINE\]NEWLINE are proved. The proof of the existence is reduced to a Wiener-Hopf equation with zero index. The solvability of the equation and hence of the problem itself follows from the uniqueness of the solution.
scientific article