The Gellerstedt problem for systems of equations of mixed type (Q1405702)

The following system \[ LU\equiv K(y) U_{xx}+ U_{yy}+ A(x,y) U_x+ B(x,y) U_y+ C(x,y)U= 0 \] is under consideration in the domain \(D\) bounded by a simple Jordan curve \(\Gamma\) lying in the halfplane \(y> 0\) with ends at points \(A_1(a_1, 0)\) and \(A_2(a_2,0)\), \(a_1< a_2\), the characteristics \(A_1C_1\), \(C_1E\), \(EC_2\), \(C_2A_2\) of the system for \(y< 0\), where \(E(e,0)\), \(a_1< e< a_2\), \(C_1({a_1+ e\over 2},y_{c_1})\), \(y_{c_1}< 0\) and \(C_2({a_2+ e\over 2}, y_{c_2})\), \(y_{2_2}< 0\). \(K(y)\), \(A(x,y)\) and \(B(x,y)\) are given numerical functions, \(yK(y)> 0\) for \(y\neq 0\), \(C(x,y)= (C_{ik}(x,y))\), \(i,k= 1,\dots, n\) is a square matrix, \(U= (u_1,u_2,\dots, u_n)\), \(n\geq 2\). Denoting \(D_+= D\cap\{y> 0\}\), \(D_1= D\cap\{y< 0\wedge x< e\}\) and \(D_2= D\cap\{y< 0\wedge x> e\}\), it is assumed that \[ K(y)\in C(\overline D_i)\wedge C^2(\overline D_i\setminus\overline{EA_i}),\;C_{jk}(x,y)\in C(\overline D_+)\wedge C(\overline D_i),\;j,k= 1,\dots, n, \] \[ A(x,y),\;B(x,y)\in C^1(\overline D_+)\wedge C(\overline D_i)\wedge C^1(\overline D_i\setminus \overline{EA_i}),\quad i= 1,2. \] The Gellerstedt problem (Problem G) is considered for the above system in the domain \(D\): Find a function \(U(x,y)\) which satisfies the conditions \[ U(x,y)\in C(\overline D)\wedge C^1(D)\wedge C^2(D_+\cup D_1\cup D_2),\;LU(x,y)\equiv 0,\;(x,y)\in D_+\cup D_1\cup D_2, \] \[ U(x,y)= \Phi(x,y),\;(x,y)\in\overline\Gamma,\;U(x,y)= \Psi(x,y),\;(x,y)\in\overline{A_1C_1}\cup \overline{A_2C_2}, \] where \(\Phi\) and \(\Psi\) are given sufficiently smooth vector functions, \(\Phi(A_1)= \Psi)A_1)\) and \(\Phi(A_2)= \Psi(A_2)\). The extremal properties of the module \[ | U(x,y)|= \sqrt{\sum^n_{i=1} u^2_i(x,y)} \] of solutions of the Problem G for a considered system are established in the domains of ellipticity, hyperbolicity, and on the whole in the mixed domain \(D\). The applications of these properties to the investigation of Problem G are given. It is demonstrated the validity of the theory outlined above for some equations which were studied in [\textit{E. I. Moiseev}, Equations of mixed type with spectral parameter, Moscow Univ. Press (1988; Zbl 0653.35003)] and [\textit{S. M. Ponomarev}, Spectral theory of the basic boundary value problem for the Lavrent'yev-Bitsadze equation of mixed type, Dr. Sci. Thesis, Moscow (1981)].