On the regularity and non-regularity of elliptic and parabolic systems (original) (raw)

In our lecture we want to answer some questions connected with the properties of weak solutions of elliptic and parabolic systems, both the linear and the quasllinear ones. We are going to present our results in the context of some known facts about the regularity and non-regularity. 1.ELLIPTIC SYSTEIYIS. In what follows we denote as m the number of equations, n-the number of variables. So u (x)= £u (x 1 t«««fXj)-•••> u m (x«,...,x }~] • The Latin indices i,J run from 1 to m, Greek ones oc , Q> from 1 to n. Denote further D^ = 'd/Bx^ # Throughout the paper, the coefficient matrices A = JA.J.W Bie supposed to be symmetric, i.e., A i1 = A li * * n * ne wn°le text, Einstein summation convention is used. The linear elliptic system with bounded and measurable coefficients on the open subset .SLcR n is of the form (° D «(A tj(x > D ft ud) * °> i=1f...tni f <2) A^CLjil),