Fractional regularisation of the Cauchy problem for Laplace's equation and application in some free boundary value problems (Q6657186)

This article discusses the regularisation of the Laplace operator through the standard equation \(\Delta u=0\) in a domain \(\Omega\times (0, \ell)\) subject to a mixture of Dirichlet or Neumann boundary conditions. The regularisation is done through the fractional derivatives in variopus settings and leads to free boundary (or obstacle) problems. The convergence of the proposed method is proven by using the Newton's method under the assumption of the invariance on the forward operator.