\(L^{2,\lambda}\) regularity of the spatial derivatives of the solutions to parabolic systems in divergence form
Under the suitable hypothesis, spatial derivatives \(D_ i v\) of the solution \(v\) to the Cauchy-Dirichlet problem for the system \[ - \sum_{i,j=1}^ n D^ i (A_{ij} (x,t) D_ j v)+ {{\partial v} \over {\partial t}}=- \sum D_ i f^ i+ f^ 0 \qquad \text{in }Q \] with the conditions \(v=u\) on the parabolic boundary \(\Gamma\) belong to the Morrey space \(L^{2,\lambda} (Q)\). It is supposed that \(A_{ij}\in C^ o (\overline{Q})\), \(f^ i\in L^{2,\lambda} (Q)\), \(0<\lambda< n+2\), \(u\in H_{-T}^{*0, \lambda/2} (Q)\cap L^{2,\lambda} (Q)\), \(D_ i u\in L^{2,\lambda} (Q)\). The estimate of \(\| u\|_{L^{2,\lambda} (Q)}\) is given.
scientific article; zbMATH DE number 495903