On the Cauchy-Dirichlet problem for fully nonlinear equations with fractional time derivative

We consider quite general fully nonlinear mixed Cauchy-Dirichlet problems with a Caputo derivative D-alpha with respect to the time variable and a in (0, 2). Under natural conditions, we show the existence of a local solution u such that D(alpha)u and the second order space derivatives D-xi,D-x j u belong to the class C-alpha theta/2,C-theta ([0, T] x (Omega) over bar), for some T positive, with theta is an element of(0, 1). Moreover, we show the uniqueness of global solutions in the same class of functions.