Weak interior second order derivative estimates for degenerate nonlinear elliptic equations (Q1314617)

Let \(L_\omega(u)\) denote a family of linear (possibly degenerate!) elliptic operators with smooth coefficients and \(L_\omega(1)\leq 0\), parametrized by \(\omega\in A\) compact, and consider the Bellman-equation \[ F(D^2 u, Du, u, x):= \min_{\omega\in A} \{L_\omega(u(x))+ f_\omega(x)\}= 0. \] The problem is to estimate the second derivative of smooth solutions \(u\) in terms of these quantities at the boundary. The author presents a complete analytic proof of this aim (in contrast to the well-known probabilistic techniques of the author) under a rather general assumption, which is close to necessity. In fact, it is shown that interior estimates may be given, where the second normal derivative at the boundary enters only with a small factor on the right-hand side.