Initial-Mixed Boundary Value Problems for Parabolic Complex Equations of Second Order with Measurable Coefficients

Abstract

In [2], the authors introduced some results on initial-boundary value problems for divergence parabolic equations of second order with measurable coefficients. In [1], the authors considered the first boundary value problem for non-divergence parabolic equations of second order with discontinuous coefficients. In this paper, we consider initial-mixed boundary value problems for non-divergence parabolic complex equations of second order in a multiply connected domain. Firstly, we give a priori estimates of solutions of the above initial-boundary value problems by the method of symmetric extension, and then by using these estimates, the methods of auxiliary functions and parameter extension, we prove the solvability for the foregoing problems. Here the condition (1.3) is weaker than the corresponding one in [1] and [3], i.e. the constant 4/3 in [1] and [3] is replaced by 3/2 in (1.3).