Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators (Q512514)

The authors obtain sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for the following second-order elliptic operator differential equations on a semiaxis: \[ \begin{cases} -u''(t)+p(t) A^2u(t)+A_1u'(t)+A_2u(t)=f(t), \quad t\in {\mathbb R_+},\\ u'(0)=K u(t), \end{cases} \] where \(f\in L_2({\mathbb R_+}; H),\) \( A, A_1\) and \(A_2\) are linear operators. The boundary condition at zero involves an abstract linear operator \(K\). Using the properties of the operator coefficients, solvability conditions are established. The norms of the intermediate derivative operators, which are closely related to the solvability conditions, are estimated.