Some regularity results for nonlinear degenerate elliptic equations (Q1343057)

The author studies regularity of solutions of the homogeneous Dirichlet problem for nonlinear partial differential operators of the form \(Au = - \text{div} (a (x,u, Du))\). A is not uniformly elliptic, but satisfies the condition \(a(x,s,\xi) \xi \geq \nu (x) |\xi |^p\), where \(\nu\) is a nonnegative function in \(L^1 (\Omega)\), \(p > 1\). The main results of the paper state that any solution of the equation \(Au = f\) lies in \(L^{\widehat p} (\Omega)\), where \(\widehat p\) depends on \(p,f\), and the dimension \(n\).