Homothetic Hodge$-$de Rham Theory and a Geometric Regularization of Elliptic Boundary Value Problems

Abstract

We introduce a homothetic extension of classical Weyl integrable geometry by generalizing the usual linear gauge transformations to affine homothetic transformations centered at a distinguished harmonic, scale-invariant form $α_d$. After relinearizing these affine gauge transformations via a suitable shift of variables, we obtain a twisted exterior calculus that is structurally equivalent to the Witten deformation of the de Rham complex. On this basis, we develop a corresponding homothetic Hodge theory: we define a twisted adjoint and homothetic Laplacian, and prove a homothetic Hodge decomposition theorem on compact Riemannian manifolds. In the context of partial differential equations, we show that the scalar homothetic Laplacian provides a rigorous diffuse interface (volume penalization) representation of elliptic boundary value problems. Modeling the Weyl scale field as a fixed distribution localized near a hypersurface, the resulting lower order geometric terms form a penalty layer that enforces Dirichlet, Neumann, or Cauchy data within a single geometric equation. This formulation yields consistent weak solutions even in the presence of classically incompatible Cauchy data. As an application, we construct a nonsingular model for point sources in elliptic field equations, which preserves the correct Coulombian far field while removing the core singularity and yielding finite field energy.