Compactness methods for certain degenerate elliptic systems (Q1313527)

The author provides a new proof of \(C^{1, \alpha}\) regularity of vector minimizers of the functional \[ I_ p (v)= \int_ \Omega |\nabla u|^ p dx, \qquad p\in (1,\infty). \] The proof consists in thorough study of estimates for regularized functionals \[ I_ p^ \varepsilon= \int_ \Omega (\varepsilon+ | \nabla u|^ 2)^{p/2} dx. \] The points with degenerate and nondegenerate gradient are studied separately using the following ideas: if \(| \nabla u|\) is small enough on a part of a ball \(B_ r\) then it is small on \(B_{r/2}\); i.e. \(u\) is more degenerate, if \(| \nabla u|\) is away from zero on a large portion of \(B_ r\) then \(u\) is well approximated by vectors of nondegenerate harmonic functions.