Harnack's principle

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Harnack Inequality for $f$-Mean Curvature Flow Open

Li, Xiang-Dong, Yan Qi · 2025

In this paper, we prove a Li-Yau-Hamilton type Harnack estimate for the $f$-mean curvature flow in Euclidean space, which can be viewed as a gradient flow of the weighed area functional with the measure density function $e^{-f}$.

Harnack Inequality for $f$-Mean Curvature Flow Open

Li, Xiang-Dong, Yan Qi · 2025

In this paper, we prove a Li-Yau-Hamilton type Harnack estimate for the $f$-mean curvature flow in Euclidean space, which can be viewed as a gradient flow of the weighed area functional with the measure density function $e^{-f}$.

Quantitative Regularity for Non-Local Operators on Metric Measure Spaces Open

SÉRGIO DE ANDRADE, PAULO · 2025

This paper develops a quantitative regularity theory for weak solutions to a class of non-local integro-differential equations on metric measure spaces. We consider operators analogous to the fractional Laplacian in a general setting where…

Nonlocal Regularity Theory for Degenerate Elliptic Equations with Rough Potentials Open

SÉRGIO DE ANDRADE, PAULO · 2025

This paper develops a regularity theory for weak solutions to a class of nonlocal degenerate elliptic equations involving rough potentials. We consider equations driven by the fractional p-Laplacian operator, perturbed by a potential term …

Nonlocal Regularity Theory for Degenerate Elliptic Equations with Rough Potentials Open

SÉRGIO DE ANDRADE, PAULO · 2025

This paper develops a regularity theory for weak solutions to a class of nonlocal degenerate elliptic equations involving rough potentials. We consider equations driven by the fractional p-Laplacian operator, perturbed by a potential term …

Quantitative Regularity for Non-Local Operators on Metric Measure Spaces Open

SÉRGIO DE ANDRADE, PAULO · 2025

This paper develops a quantitative regularity theory for weak solutions to a class of non-local integro-differential equations on metric measure spaces. We consider operators analogous to the fractional Laplacian in a general setting where…

Quantitative Regularity for Non-Local Operators on Metric Measure Spaces Open

SÉRGIO DE ANDRADE, PAULO · 2025

This paper develops a quantitative regularity theory for weak solutions to a class of non-local integro-differential equations on metric measure spaces. We consider operators analogous to the fractional Laplacian in a general setting where…

Nonlocal Regularity Theory for Degenerate Elliptic Equations with Rough Potentials Open

SÉRGIO DE ANDRADE, PAULO · 2025

This paper develops a regularity theory for weak solutions to a class of nonlocal degenerate elliptic equations involving rough potentials. We consider equations driven by the fractional p-Laplacian operator, perturbed by a potential term …

Towards a characterization of elliptic Harnack inequality for jump processes Open

Malmquist, Jens · 2025

Let $X$ be an isotropic unimodal Lévy jump process on $\mathbb{R}^d$. We develop probabilistic methods which in many cases allow us to determine whether $X$ satisfies the elliptic Harnack inequality (EHI), by looking only at the jump kerne…

Towards a characterization of elliptic Harnack inequality for jump processes Open

Malmquist, Jens · 2025

Let $X$ be an isotropic unimodal Lévy jump process on $\mathbb{R}^d$. We develop probabilistic methods which in many cases allow us to determine whether $X$ satisfies the elliptic Harnack inequality (EHI), by looking only at the jump kerne…

Elliptic Harnack inequality and its applications on Finsler metric measure spaces Open

Xinyue Cheng, Liulin Liu, Yuxin Zhang · 2025

STABILITY RESULTS FOR NONLOCAL SERRIN-TYPE PROBLEMS, ANTISYMMETRIC HARNACK INEQUALITIES AND GEOMETRIC ESTIMATES Open

J. C. Thompson · 2025

New Boundary Harnack Principles for Divergence Form Elliptic Equations with Right Hand Side Open

Xiaoyan Liang, Y. Cao · 2025

We establish new boundary Harnack principles for divergence form equations with right hand side. In Lipschitz domains, we prove new boundary Harnack principles when the right hand side exhibits polynomial decay. Moreover, in Hölder domains…

Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term Open

Domenico Vuono · 2025

We consider weak solutions of the equation $$\begin{aligned} -\Delta _p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text {in } \Omega , \end{aligned}$$ where H is in some cases called Finsler norm, $$\Omega…

Li-Yau estimates and Harnack inequalities for nonlinear slow diffusion equations on a smooth metric measure space Open

Ali Taheri, Vahideh Vahidifar · 2025

Time‐insensitive nonlocal parabolic Harnack estimates Open

Naian Liao, Marvin Weidner · 2025

We establish new Harnack estimates that defy the waiting‐time phenomenon for global solutions to nonlocal parabolic equations. Our technique allows us to consider general nonlocal operators with bounded measurable coefficients. Moreover, w…

Harnack Inequality for Self-Repelling Diffusions Driven by Subordinated Brownian Motion Open

Yaqin Sun, Litan Yan · 2025

In this paper, we consider a self-repelling diffusion driven by the Lévy process. By using the coupling argument, we establish the corresponding Bismut formula and Harnack inequality.

Path-Distribution Dependent SDEs: Well-Posedness and Asymptotic Log-Harnack Inequality Open

Feng‐Yu Wang, Chenggui Yuan, Xiaoyu Zhao · 2025

We consider stochastic differential equations on $\mathbb R^d$ with coefficients depending on the path and distribution for the whole history. Under a local integrability condition on the time-spatial singular drift, the well-posedness and…

Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions Open

Junyoung Park · 2025

The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean con…

Elliptic Harnack inequality and its applications on Finsler metric measure spaces Open

Xinyue Cheng, Liulin Liu, Yuxin Zhang · 2025

In this paper, we study the elliptic Harnack inequality and its applications on forward complete Finsler metric measure spaces under the conditions that the weighted Ricci curvature ${\rm Ric}_{\infty}$ has non-positive lower bound and the…

A Strengthening of the Harnack Inequality Open

Marek Svetlik · 2025

We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.

Path-distribution dependent SDEs: Well-posedness and asymptotic log-Harnack inequality Open

Feng‐Yu Wang, Chenggui Yuan, Xiaoyu Zhao · 2025

Boundary Harnack principle for jump diffusions Open

Zhen-Qing Chen · 2025

Differential Harnack estimates for the semilinear parabolic equation with three exponents on $ \mathbb{R}^{n} $ Open

Fanqi Zeng, W. T. Geng, Ke An Liu, Boya Wang · 2025

In this paper, we thought about the positive solutions to the semilinear parabolic equation with three exponents, and obtained several differential Harnack estimates of the positive solutions to the equation. As applications of the main th…

The Boundary Harnack Principle and the 3G Principle in Fractal-Type Spaces Open

Anthony Graves‐McCleary, Laurent Saloff‐Coste · 2024

We prove a generalized version of the $3G$ Principle for Green's functions on bounded inner uniform domains in a wide class of Dirichlet spaces. In particular, our results apply to higher-dimensional fractals such as Sierpinski carpets in …

A boundary Harnack principle and its application to analyticity of 3D Brownian intersection exponents Open

Yifan Gao, Xinyi Li, Yifan Li, Liu Run-sheng, Xiaolong Liu · 2024

We show that a domain in $\mathbb{R}^3$ with the trace of a 3D Brownian motion removed almost surely satisfies the boundary Harnack principle (BHP). Then, we use it to prove that the intersection exponents for 3D Brownian motion are analyt…

Local boundedness and Harnack inequality for an inverse variational inequality problem with double nonlinear parabolic operator in finance Open

Jia Li, Zhipeng Tong · 2024

The objective of this paper is to investigate a class of initial boundary value problems for inverse variational inequalities that arise from financial matters. By utilizing the energy inequality on a localized cylindrical region and the C…

The nonlocal Harnack inequality for antisymmetric functions: an approach via Bochner's relation and harmonic analysis Open

Serena Dipierro, Mateusz Kwaśnicki, J. C. Thompson, Enrico Valdinoci · 2024

We revisit a Harnack inequality for antisymmetric functions that has been recently established for the fractional Laplacian and we extend it to more general nonlocal elliptic operators. The new approach to deal with these problems that we …

The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth Open

Simone Ciani, Eurica Henriques, Igor I. Skrypnik · 2024

In this work we prove that the non-negative functions u ∈ L loc s ⁢ ( Ω ) {u\in L^{s}_{\rm loc}(\Omega)} , for some s > 0 {s>0} , belonging to the De Giorgi classes ⨍ B r ⁢ ( 1 - σ ) ⁢ ( x 0 ) | ∇ ( u - k …

A generalisation of simple Harnack curves Open

Lionel Lang · 2024

In this paper, we suggest the following generalisation of Mikhalkin’s simple Harnack curves: a generalised simple Harnack curve is a parametrised real algebraic curve in $$({\mathbb {C}}^{*})^{2}$$ with totally real logarithmic …

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