Solvability conditions for boundary-value problems for elliptic operator-differential equations with discontinuous coefficient (Q1946435)

Let \(H\) be a separable Hilbert space, and let \(A\) be a self-adjoint uniformly positive operator in~\(H\). Let \(H_{\theta}\), \(\theta\geq0\), be the scale of Hilbert operators generated by~\(A\). Let \(f\in L_2(\mathbb R_+;H)\), \(K\in L(H_{3/2},H_{1/2})\), and let \(A_1\) be a linear operator such that \(\text{Dom}(A_1)\supset\text{Dom}(A)\) and \(A_1A^{-1}\) is bounded in \(H\), while \(\rho(t)\) is a positive piecewise constant function with a single point of discontinuity. The authors state without detailed proofs several theorems giving sufficient conditions in order for the boundary value problem \[ -u''(t)+\rho(t)A^2u(t)+A_1u'(t)=f(t),\;t\in\mathbb R_+,\;u'(0)=Ku(0) \] to be regularly solvable, i.e., for each function \(f\), there is a function \(u\in W_2^2(\mathbb R_+;H)\) satisfying the given equation a.e. in \(\mathbb R_+\) and the boundary condition in the sense of convergence in the space \(H_{1/2}\), i.e., \(\lim_{t\to0}\|u'(t)-Ku(t)\|_{1/2}=0\) and the following estimate holds: \(\|u\|_{W_2^2(\mathbb R_+;H)}\leq\mathrm{const}\cdot\|f\|_{L_2(\mathbb R_+;H)}\).