Let \(\Omega\) is a bounded set in \(\mathbb{R}^d\) (\(d\geq 2\)). The authors consider the elliptic system \[ -\sum_{i,j=0}^d\sum_{k=1}^m\partial_i(a_{i,j}^{l,k}(t,x)\partial_ju_k(t,x))=0, \quad l=1,\dots,m, \quad (t,x)\in \mathbb{R}_+\times \Omega, \] subjected to the mixed Dirichlet/Neumann lateral boundary conditions \[ u=0\text{ on }\mathbb{R}_+\times \mathcal{D},\quad \nu \cdot A\nabla_{t,x}u=0\text{ on }\mathbb{R}_+\times(\partial \Omega \setminus \mathcal{D}). \] Here, \(\nu\) is the outer normal vector to \(\mathbb{R}_+\times \Omega\), \(\mathcal{D}\) is a closed subset of \(\partial \Omega\) and \(A=A(t,x):= \big(a_{i,j}^{l,k}(t,x)\big)_{i,j=0,\dots,d}^{l,k=1,\dots,m}\) is the coefficient tensor (the coefficients being complex).
The set \(\Omega\) is assumed to be bounded, open, connected, and \(d\)-regular (with respect to the Lebesgue measure), and the set \(\mathcal{D}\) is assumed either empty or \(d-1\)-regular (with respect to the Hausdorff measure). Finally, the set \(\partial \Omega \setminus \mathcal{D}\) is assumed to be locally Lipschitz.
The coefficient tensor \(A\) is assumed bounded in \(\mathbb{R}_+\times \Omega\), satisfying an ellipticity/accretivity condition, and such that \(\|A-A_0\|_C<\infty\), for some \(t\)-independent tensor \(A_0\), where \(\|\cdot\|_C\) is a modified Carleson norm.
The authors establish a priori estimates for weak solutions to the above problem satisfying one of the three following inhomogeneous boundary conditions on \(\{0\}\times \Omega\):
1) \(u_{\mid t=0}=\varphi\in L^2(\Omega)^m\),
2) \({(A\nabla_{t,x}u)_{\bot}}_{\mid t=0}=\varphi\in L^2(\Omega)^m\),
3) \((A\nabla_xu)_{\mid t=0}=\varphi\in L^2(\Omega)^m\).
The a priori estimate is proved for weak solutions whose non-tangential maximal function has finite \(L^2\)-norm in the case of the boundary conditions \(2)\) and \(3)\), and for weak solutions satisfying a Lusin area bound in the case of the boundary condition \(1)\). In both cases, additional regularity results are established when \(A=A_0\).
A well posedness result is also proved for \(t\)-independent coefficient tensors satisfying some additional condition and for \(t\)-dependent coefficient tensors sufficiently near (with respect to the above modified Carleson norm) to a \(t\)-independent coefficient tensor for which well posedness holds.
The proof relies on the transformation of the above elliptic system with the mixed lateral boundary conditions into an equivalent non-autonomous evolution first order equation and on the Kato root estimate for operators with mixed boundary conditions.