Communications in Mathematical Sciences
Volume 16 (2018)
Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization
Pages: 1395 – 1426
In this paper, we introduce new characterizations of the 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 conditions. 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 equations as constraints.
new spectral fractional Laplace operator, nonzero boundary conditions, very weak solution, finite element discretization, error estimates
2010 Mathematics Subject Classification
26A33, 35S15, 49K20, 65N12, 65N30, 65R20
The work of the first and second author is partially supported by NSF grants DMS-1521590 and DMS-1818772 and Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036.
Received 6 September 2017
Received revised 8 June 2018
Accepted 8 June 2018
Published 19 December 2018