Regularity for non-homogeneous systems

 My starting point is the analysis of the behavior of manifold-constrained minima to certain non-homogeneous functionals: under sharp assumptions, we prove that they are regular everywhere, except on a negligible, "singular" set of points. The presence of the singular set is in general unavoidable. Looking at minima as solutions to the associated Euler-Lagrange system does not help: it presents an additional component generated by the curvature of the manifold having critical growth in the gradient variable. It is then natural to consider general systems of type

\begin{flalign}\label{ab0}-\diver \ a(x,Du)=f\end{flalign}

and study how the features of $f$ and of the partial map $x\mapsto a(x,z)$ influence the regularity of solutions. In this respect, I am able to cover non-linear tensors with exponential type growth conditions as well as with unbalanced polynomial growth: I prove everywhere Lipschitz regularity for vector-valued solutions to \eqref{ab0} under optimal assumptions on forcing term and space-depending coefficients. This approach also yields optimal regularity results for obstacle problems.