Parabolic equations with BMO coefficients in Lipschitz domains
The present paper deals with certain Sobolev-type estimates for the weak solutions of the following initial-boundary value problem for a uniformly parabolic operator \[ \begin{cases} u_t-(a_{ij}u_{x_j})_{x_i}=u_t- \operatorname{div} (A\nabla u)= \operatorname{div} {\mathbf f} & \text{ in } \Omega_T,\\ u=0 & \text{ on } \partial_p\Omega_T. \end{cases} \] The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the parabolic constants of the operator. There are proved relevant \(L^p\) estimates, assuming that the coefficients are in the parabolic class of functions with bounded mean oscillation (BMO) with small enough BMO-seminorm.
scientific article; zbMATH DE number 2136114
scientific article