  In this paper we introduce two new fractional versions of the Laplacian. The
first one is based on the classical formula that writes the usual Laplacian as
the sum of the eigenvalues of the Hessian. The second one comes from looking at
the classical fractional Laplacian as the mean value (in the sphere) of the
1-dimensional fractional Laplacians in lines with directions in the sphere. To
obtain this second new fractional operator we just replace the mean value by
the mid-range of 1-dimensional fractional Laplacians with directions in the
sphere. For these two new fractional operators we prove a comparison principle
for viscosity sub and supersolutions and then we obtain existence and
uniqueness for the Dirichlet problem. We also show that solutions are
$C^\gamma$ smooth up to the boundary when the exterior datum is also H\"older
continuous. Finally, we prove that for the first operator we recover the
classical Laplacian in the limit as $s\nearrow 1$.
