Degenerate Laplacian: A Classification by Helicity

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.