On time-periodic solutions to parabolic boundary value problems (Q2423407)

The paper deals with the parabolic boundary value problem \[ \begin{cases} \partial_t u+Au=f,& \text{ in }\mathbb{R}\times \Omega,\\ B_ju=g_j, & \text{ on }\mathbb{R}\times \partial \Omega,\tag{1} \end{cases} \] where \(A\) is an elliptic operator of order \(2m\), the operators \(B_j,\,\,j=1,\dots,m\) satisfy an appropriate complementing boundary condition, and the domain \(\Omega\) is the whole space, the half-space or a bounded domain of \(\mathbb{R}^n\). The authors decompose problem \((1)\) into two problems, namely an elliptic problem in the sense of Agmon-Douglis-Nirenberg, and a purely oscillatory problem. For the second problem, they investigate the existence and uniqueness of the time-periodic solutions. In the whole-space and half-space, the authors give an explicit formula for the solution and establish coercive \(L^p\) estimates in Sobolev spaces.