  In this paper we introduce new characterizations of spectral fractional
Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary
conditions. The classical cases with homogeneous boundary conditions arise as a
special case. We apply our definition to fractional elliptic equations of order
$s \in (0,1)$ with nonzero Dirichlet and Neumann boundary condition. Here the
domain $\Omega$ is assumed to be a bounded, quasi-convex Lipschitz domain. To
impose the nonzero boundary conditions, we construct fractional harmonic
extensions of the boundary data. It is shown that solving for the fractional
harmonic extension is equivalent to solving for the standard harmonic extension
in the very-weak form. The latter result is of independent interest as well.
The remaining fractional elliptic problem (with homogeneous boundary data) can
be realized using the existing techniques. We introduce finite element
discretizations and derive discretization error estimates in natural norms,
which are confirmed by numerical experiments. We also apply our
characterizations to Dirichlet and Neumann boundary optimal control problems
with fractional elliptic equation as constraints.
