Hölder continuity of solutions to quasilinear elliptic equations involving measures (Q1335526)

This article contains sharp regularity results for solutions of nonhomogeneous quasilinear elliptic equations whose prototype is \[ -\text{div} \bigl( | \nabla u |^{p-2} \nabla u \bigr) = \mu, \] where \(\mu\) is a positive measure satisfying the condition \(\mu \bigl( B(x,r) \bigr)\leq Mr^ \rho\). It turns out that the solution \(u\) is Hölder continuous if and only if \(\rho > n - p\). The author proves that to get Hölder continuity of order \(\alpha\) it is enough to have \(\rho > n - p + \alpha (p-1)\), and that this result is sharp in the sense that there are examples of solutions which are not Hölder continuous with exponent \(\beta\) if \(\rho < n - p+ \beta (p-1)\). Extensions of these results to the variable coefficient case are also presented.