Jessica Slegers | Searcharxiv

arXiv2026

Harnack-type inequalities for nonlinear evolution equations

Harnack inequalities are useful qualitative tools for understanding the properties of partial differential equations. Originally discovered as a property of harmonic functions, Harnack inequalities have since been studied for solutions of wider classes of elliptic and parabolic problems. In this monograph, we take particular interest in deriving Harnack inequalities for solutions of nonlinear evolution equations. We focus on exploring the methods introduced by Li and Yau in the case of the linear heat equation and later extended to nonlinear problems by Auchmuty and Bao. Prior to presenting these results, we study a minimisation problem, which appears naturally in the proofs. After establishing a family of three general Harnack inequality results by Auchmuty and Bao, we investigate applications to deriving Harnack inequalities satisfied by solutions of the porous medium equation and weak solutions of the parabolic problem associated with the $p$-Laplace operator, which we refer to here as the $p$-diffusion equation. Finally, we demonstrate a common application of Harnack inequalities by proving the local space-time Hölder continuity of solutions to a class of linear evolution problems. The proof is based on methods introduced by Moser during his seminal work on Harnack inequalities during the 1960s. We conclude by suggesting potential opportunities for future work following on from the topics discussed here.

math.AP↗

arXiv2025

A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators

In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schrödinger equation on a Riemannian manifold $M$ with nonnegative Ricci curvature, where the potential term $V$ is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least $C^4$ on $M$. We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only $C^2$ on $M$. In the particular case that $V$ is the quadratic potential $V(x)=|x|^2$ and $M$ is the Euclidean space $\mathbb{R}^d$, we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schrödinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator $Δ- x\cdot \nabla$ with quadratic potential in $\mathbb{R}^d$.

math.AP↗