Degenerate Laplacian: A Classification by Helicity

TL;DR

This paper introduces a class of degenerate Laplacian operators influenced by topological constraints, analyzing their properties and solutions despite loss of ellipticity, with implications for diffusion models and particle dynamics.

Contribution

It defines and studies the properties of orthogonal Laplacians arising from degenerate metrics, establishing existence and uniqueness of solutions despite degeneracy.

Findings

Orthogonal Laplacian operators can be associated with a Sobolev-like Hilbert space.

A Poincaré-like inequality holds for the orthogonal Laplacian.

Unique weak solutions to the orthogonal Poisson equation exist under certain conditions.

Abstract

We study a class of generalized Laplacian operators by violating the ellipticity with degenerate metric tensors. The theory is motivated by the statistical mechanics of topologically constrained particles. In the context of diffusion models, the metric tensor is given by $g=-\mathcal{J}^2$ with a generalized Poisson matrix $\mathcal{J}$ that dictates particle dynamics. The standard Euclidean metric corresponds to the symplectic matrix of canonical Hamiltonian systems. However, topological constraints bring about nullity to $\mathcal{J}$, resulting in degeneracy in the corresponding diffusion operator; we call such an operator an orthogonal Laplacian (since the ellipticity is broken in the direction parallel to the nullity), and denote it by $\Delta_{\perp}$. Although all nice properties pertinent to the ellipticity are generally lost for $\Delta_{\perp}$, a finite helicity of $\mathcal{J}$ helps to recover some of them by preventing foliation of space. We show that $-\left(\Delta_{\perp}u,v\right)$ defines an inner product of a Sobolev-like Hilbert space, and satisfies a Poincar\'e-like inequality $-\left(\Delta_{\perp}u,u\right)\geq C\lvert\lvert{u}\rvert\rvert^{2}_{L^2}$. Applying Riesz's representation theorem, we obtain a unique weak solution of the orthogonal Poisson equation.