Regularity of weak solutions to the model Venttsel problem for linear parabolic systems with nonsmooth in time principal matrix: \(A(t)\)-caloric approximation method (Q728475)

The author considers the model Venttsel-type problem for linear parabolic systems of equations \[ \begin{cases} u_t-\text{div }(a(z)\nabla u)=f(z), & z \in Q_1^+(0),\\ u_t-\text{div' }(b(z')\nabla'u)+\frac{\partial u}{\partial \nu_a}=\psi(z'), & z'\in \Gamma_1(0) \end{cases} \] where \(z=(x,t),\) \(z'=(x',0,t).\) The Venttsel-type boundary condition is fixed on the flat part of the lateral surface of a given cylinder. It is defined by a parabolic operator (with respect to the tangential derivatives) and the conormal derivative. The Hölder continuity of every weak solution of the problem is proved under optimal assumptions on the data. In particular, only boundedness in the time variable of the principal matrices of the system and the boundary operator are assumed. All results are obtained by so-called \(A(t)\)-caloric approximation method.