Distribution-transported boundary integral regularization for physically constrained cauchy data completion in the bi-laplace equation

This paper presents a structure-preserving regularization framework for Cauchy data completion problems governed by the two-dimensional bi-Laplace equation. On a boundary partitioned into an accessible segment carrying noisy physical measurements and an inaccessible segment with unknown traces, inverse continuation is severely ill-posed. To resolve this, we first introduce a modified indirect boundary integral formulation driven by two potential densities. This formulation establishes a well-defined linear operator system for the complete quadruplet of biharmonic boundary traces, ensuring consistent structural coupling across operators. To address physical admissibility, our second contribution introduces a distribution-transported Tikhonov regularization scheme for missing boundary traces subject to known lower and upper physical bounds. By parametrizing the constrained trace via a smooth cumulative distribution function acting on an unconstrained latent state, prescribed physical ranges are enforced directly by construction, avoiding non-smooth projection or post-processing clipping. We rigorously prove the existence, stability, convergence, and regularization consistency of the resulting nonlinearly parametrized estimates. Numerical experiments on the unit disk compare classical Tikhonov regularization against transported variants utilizing logistic, normal, exponential, and new XLindley cumulative distribution maps. The findings demonstrate that while standard Tikhonov regularization yields unphysical undershoots, the transported framework guarantees strict physical bound adherence without sacrificing reconstruction fidelity or residual convergence.