Boundary regularity results for weak solutions of subquadratic elliptic systems
Lisa Beck · 2008
This thesis makes a contribution to the field of regularity theory of second-order nonlinear elliptic systems of partial differential equations. We consider weak solutions of the inhomogeneous elliptic system - div a(x,u,Du) = b(x,u,Du) on a bounded (and sufficiently regular) domain with prescribed boundary data. The coefficients are assumed to satisfy standard subquadratic continuity, ellipticity and growth conditions, and the inhomogeneity obeys either a controllable or a natural growth condition. Under these assumptions, the following higher integrability and regularity results (up to the boundary) are achieved: * Characterization of regular points for Du; * Calderón-Zygmund estimates; * Partial Hölder continuity of u outside a singular set of Hausdorff dimension less than n-p, provided that the low dimensional assumption n ∈ (p,p+2] holds; * Reduction of the Hausdorff dimension of the singular set of Du, in particular, we give conditions which guarantee the existence of regular boundary points for Du.