arXiv subjects

Vahideh Vahidifar

Publications and source records attributed to Vahideh Vahidifar.

8 recordsLinked to original sources

arXiv2025

The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications

This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation (NPME) in the context of smooth metric measure spaces. The diffusion operator here is the f-Laplacian and the gradient estimates of interest are mainly of Hamilton-Souplet-Zhang types. These estimates are established using a variety of methods and techniques and several implications, most notably, to parabolic Liouville-type results and characterisation of ancient solutions are given. The problem is posed in the general framework where the metric and potential evolve with time and the proofs make use of natural lower bounds on the time derivative of the metric and the Bakry-Émery m-Ricci curvature tensors. Our results extend and improve various existing ones in the literature.

math.AP↗

arXiv2025

Li-Yau Estimates and Harnack Inequalities for Nonlinear Slow Diffusion Equations on a Smooth Metric Measure Space

We present new gradient estimates and Harnack inequalities for positive solutions to nonlinear slow diffusion equations. The framework is that of a smooth metric measure space $(\mathscr M,g,dμ)$ with invariant weighted measure $dμ=e^{-ϕ} dv_g$ and diffusion operator $Δ_ϕ=e^ϕ{\rm div} (e^{-ϕ} \nabla)$ -- the $ϕ$-Laplacian. The nonlinear slow diffusion equation, then, for $x \in {\mathscr M}$ and $t>0$, and fixed exponent $p>1$, takes the form \begin{equation*} \partial_t u (x,t) - Δ_ϕu^p (x,t) = \mathscr N (t,x,u(x,t)). \end{equation*} We assume that the metric tensor $g$ and potential $ϕ$ are space-time dependent; hence the same is true of the usual metric and potential dependent differential operators and curvature tensors. The estimates are established under natural lower bounds on the Bakry-Émery $m$-Ricci curvature tensor and the time derivative of metric tensor. The curious interplay between geometry, nonlinearity and evolution and their influence on the estimates is at the centre of this investigation. The results here considerably extend and improve earlier results on slow diffusion equations. Several implication, special cases and corollaries are presented and discussed.

math.AP↗

arXiv2025

The nonlinear fast diffusion equation on smooth metric measure spaces: Hamilton-Souplet-Zhang estimates and a Ricci-Perelman super flow

This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the $f$-Laplacian. The gradient estimates of interest are mainly of Hamilton-Souplet-Zhang or elliptic type and are proved using different set of methods and techniques. Various implications notably to parabolic Liouville type results and characterisation of ancient solutions are given. The problem is considered in the general setting where the metric and potential evolve under a super flow involving the Bakry-Émery $m$-Ricci curvature tensor. The remarkable interplay between geometry, nonlinearity, and evolution -- and their intricate roles in the estimates and the maximum exponent range of fast diffusion -- is at the core of the investigation.

math.AP↗

arXiv2024

Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general elliptic and parabolic Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.

math.AP↗

arXiv2023

Souplet-Zhang and Hamilton type gradient estimates for nonlinear elliptic equations on smooth metric measure spaces

In this article we present new gradient estimates for positive solutions to a class of nonlinear elliptic equations involving the f-Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet-Zhang and Hamilton types respectively and are established under natural lower bounds on the generalised Bakry-Émery Ricci curvature tensor. From these estimates we derive amongst other things Harnack inequalities and general global constancy and Liouville-type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.

math.AP↗

arXiv2023

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to a nonlinear parabolic equation involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman-Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry-Émery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed.

math.AP↗

arXiv2023

Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications

This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving general Harnack inequalities and Liouville-type theorems to mention a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some important consequences are presented and discussed.

math.AP↗

arXiv2022

On Multiple Solutions to a Family of Nonlinear Elliptic Systems in Divergence Form Coupled with an Incompressibility Constraint

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll} \dive\{\A(|x|,|u|^2,|\nabla u|^2) \nabla u\} + \B(|x|,|u|^2,|\nabla u|^2) u = \dive \{ \mcP(x) [{\rm cof}\,\nabla u] \} \quad &\text{ in} \ Ω, \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ Ω, \\ u =φ\ &\text{ on} \ \partial Ω, \end{array} \right. \end{align*} where $Ω\subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $φ$ is a prescribed boundary condition. Moreover $\mathscr{P}$ is a hydrostatic pressure associated with the constraint $\det \nabla u \equiv 1$ and $\A = \A(|x|,|u|^2,|\nabla u|^2)$, $\B = \B(|x|,|u|^2,|\nabla u|^2)$ are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draws upon intimate links with the Lie group ${\bf SO}(n)$, its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably a discriminant type quantity $Δ=Δ(\A,\B)$, prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.

math.AP↗