A boundary value problem for elliptic differential-operator equations (Q1572962)

The basic part of this article deals with boundary value problems of the type \[ -u''(x)+ Au(x)= f(x),\quad -(A_{\nu 0}u(x))''+ A_{\nu 1}u(x)= f_\nu(x),\quad \nu= 1,\dots, s, \] \[ \alpha_k u^{(p_k)}(0)+ \beta_ku^{(p_k)}u(1)+ \sum^{N_k}_{j= 1} \delta_{kj} u(x_{kj})= 0,\quad k= 1,2, \] with \(x\in [0,1]\), \(p_k\in \{0,1\}\), \(x_{kj}\in [0,1]\); \(A\) is an operator in a Hilbert space \(H\), \(A_{\nu 0}\) and \(A_{\nu 1}\) are operators between the Hilbert space \(H\) and some Hilbert spaces \(H_1,\dots, H_s\), \(f(\cdot): [0,1]\to H\), \(f_\nu(\cdot): [0,1]\to H_\nu\), \(u(\cdot): [0,1]\to H\) is an unknown function, and similar ``principally'' boundary value problems for functions of two variables \[ -D^2_x u(x,y)- a(y) D^{2m}_y u(x,y)+ Bu(x,\cdot)|_y= f(x,y), \] \[ \alpha_k D^{p_k}_x u(0, y)+ \beta_k D^{p_k}_x u(1,y)+ \sum^{N_k}_{j= 1}\delta_{kj}(x_{kj}, y)= 0,\quad k= 1,2, \] \[ D^2_x\Biggl(\gamma_\nu D^{m_\nu}_y u(x,0)+ \eta_\nu D^{m_\nu}_y u(x, i)+ \sum^{Q_\nu}_{j= 1} \mu\nu j D^{m_\nu}_y u(x, y_{\nu j})+ T_\nu u(x,\cdot)\Biggr)+ T_{\nu 0} u(x,\cdot)= f_\nu(x), \] \[ x\in [0,1],\quad \nu= 1,\dots, s, \] \[ \gamma_\nu D^{m_\nu}_y u(x,0)= \eta_\nu D^{m_\nu}_y u(x, i)+ \sum^{Q_\nu}_{j= 1} \mu\nu j D^{m_\nu}_y u(x, y_{\nu j})+ T_\nu u(x,\cdot)= 0, \] \[ x\in [0,1],\quad \nu= s+ 1,\dots, 2s. \] The author formulates theorems on conditions under which the operator \(\mathbb{L}\) generated by these boundary value problems is Fredholm (he states that these systems are not overdetermined and do not satisfy the Lopatinskii condition). In the appendix some particular cases and modifications are considered.