Estimates of the Simple Layer Potential for Parabolic Systems in the Dini Space
The article considers a simple layer potential which has at its kernel the fundamental matrix of solutions to the Petrovsky parabolic second-order system with one spatial variable x. It is assumed that the coefficients of the system are bounded and uniformly continuous, with their modulus of continuity satisfying the double Dini condition. The potential is considered in a semibounded (with respect to x) curvilinear domain with a nonsmooth lateral boundary belonging to the Dini-Hölder class. The density of the potential belongs to the Dini space. Estimates for the second order spatial derivative of this potential in the Dini class are established. The obtained estimates describe the possible growth of this derivative in approaching the domain lateral boundary.