The paper focuses on the Lp-Positivity Preservation property (Lp-PP for short) on a Riemannian manifold (M,g). It states that any Lp function u with 1<p<+∞, which solves (−Δ+1)u≥0 on M in the sense of distributions must be non-negative. Our main result is that the Lp-PP holds if (the possibly incomplete) M has a finite number of ends with respect to some compact domain, each of which is q-parabolic for some, possibly different, values 2p/(p−1)<q≤+∞. When p=2, since ∞-parabolicity coincides with geodesic completeness, our result settles in the affirmative a conjecture by M. Braverman, O. Milatovic and M. Shubin in 2002. On the other hand, we also show that the Lp-PP is stable by removing from a complete manifold a possibly singular set with Hausdorff co-dimension strictly larger than 2p/(p−1) or with a uniform Minkowski-type upper estimate of order 2p/(p−1). The threshold value 2p/(p−1) is sharp as we show that when the Hausdorff co-dimension of the removed set is strictly smaller, then the Lp-PP fails. This gives a rather complete picture. The tools developed to carry out our investigations include smooth monotonic approximation and consequent regularity results for subharmonic distributions, a manifold version of the Brezis–Kato inequality, Liouville-type theorems in low regularity, removable singularities results for Lp-subharmonic distributions and a Frostman-type lemma. Since the seminal works by T. Kato, the Lp-PP has been linked to the spectral theory of Schrödinger operators with singular potentials Δ−V. Here we present some applications of the main results of this paper to the case where V∈Llocp, addressing the essential self-adjointness of the operator when p=2 and whether or not Cc∞(M) is an operator core for Δ−V in Lp.

We study a novel degenerate and singular elliptic operator Δ˜(τ,χ) defined by Δ˜(τ,χ)u=τ(x,Du)(|Du|Δ1u+χ(x,Du)Δ∞u), where the singular weights τ(x,s)>0 and χ(x,s)≥0 are continuous functions on Ω×Rn∖{0}. The operator Δ˜(τ,χ) is an extension of Δ(p,q)u=|Du|qΔ1u+(p−1)|Du|p−2Δ∞u,p≥1,q≥0, introduced by the authors in Baravdishet al. (2020), which in turn is an extension of the p-Laplace operator Δp. We establish the well-posedness of the Neumann boundary value problem for the parabolic equation ut=Δ˜(τ,χ)u in the framework of viscosity solutions. For the solution u, the weight χ controls the evolution along the tangential and the normal directions, respectively, on the level surface of u. The weight τ controls the total speed of the evolution of u. We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator Δ˜(τ,χ) gives better results than both the Perona–Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.