Partial Regularity Of Quasi-linear Elliptic Systems With Discontinuous Coefficients And Green Function Of A-harmonic Operators

The present dissertation mainly is composed of two part:In the first section,I mainly establish the partial regularity with a sharp H(?)lder exponent of every weak solutions for a class of quasi-linear elliptic systems,where the coefficient operator satisfies VMO condition in variable x uniformly with respect to u and the lower order item satisfies the natural growth condition.When the coefficient function in variable x is bounded and measurable this work improve the uncertainty of H(?)lder exponent,where the the H(?)lder continuity is obtained using the De Giorgi-Moser-Nash reiteration method.Our result on the condition that quasi-linear coefficient satisfies the VMO is consistent with that of continued coefficient equations,which illuminates that VMO is the substitute for the "continuity" in the integrable function space;In the second section we investigate the concept for the other class of Green function of divergence degenerate elliptic operator in the sense of distribution,and with the modified Green functions method some local estimates and the comparison with p-Laplacian operator for the Green function are obtained.It is of great significance theoretically that the Green function act as a core function in the hole-filling technique so that this comparison is necessary(reference to the dissertation of Lu,which have included the detail content and will not be given here).