Partial Hölder continuity results for solutions of nonlinear non variational elliptic systems with strictly controlled growth (Q1589923)

The authors study partial regularity results for strong solutions \(u\in H^2(\Omega, \mathbb{R}^N)\) of the fully nonlinear nonvariational system \(a(x, u,Du,D^2u)= b(x,uDu)\), \(x\in \Omega\). Here, \(\Omega\subset \mathbb{R}^n\), \(n> 4\), is a bounded open set, \(a\), \(b\) are \(\mathbb{R}^N\)-valued functions. A number of technical conditions are imposed on \(a\) and \(b\), the most important of which may be summarized as follows: An ellipticity condition on \(a\), which further implies linear growth w.r.t \(D^2u\) and that the linearization of \(u\) deviates not too much from the Laplace operator. Measurability w.r.t \(x\), continuity w.r.t \(u\) and \(Du\) and continuous differentiability w.r.t \(D^2u\). For the lower-order term \(b\), subcritical growth (w.r.t to the linear growth of \(a)\) and the solution class \(H^2\) is assumed. Under these conditions, Hölder regularity of \(Du\) is shown, possibly except on a set of measure \(0\). Related results can be found in [\textit{S. Campanato}, Rend. Mat. Appl., VII. Ser. 10, No. 3, 531-549 (1990; Zbl 0777.35028)].