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-\'Emery 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.Comment: 41 page

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

This article presents new elliptic gradient estimates for positive solutions to nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-\'Emery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville type theorems to name 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 consequences are presented and discussed

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} \ \Omega , \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $\varphi$ 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 \Delta=\Delta(\A,\B), prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.Comment: 24 page

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-\'Emery 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

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} \ \Omega , \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $\varphi$ 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 \Delta=\Delta(\A,\B), prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions

Differential Harnack estimates for a weighted nonlinear parabolic equation under a super Perelman-Ricci flow and implications

In this paper we derive new differential Harnack estimates of Li-Yau type for positive smooth solutions to a class of nonlinear parabolic equations in the form Laϕ[w]:=[∂∂t−a(x,t)−Δϕ]w(x,t)=G(t,x,w(x,t)),t>0,on smooth metric measure spaces where the metric and potential are time dependent and evolve under a $({\mathsf k}, m)$-super Perelman-Ricci flow. A number of consequences, most notably, a parabolic Harnack inequality, a class of Hamilton type global curvature-free estimates and a general Liouville type theorem together with some consequences are established. Some special cases are presented to illustrate the strength of the results.</p

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-\'Emery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.</p

Differential Harnack estimates for a weighted nonlinear parabolic equation under a super Perelman-Ricci flow and implications

In this paper we derive new differential Harnack estimates of Li-Yau type for positive smooth solutions to a class of nonlinear parabolic equations in the form \begin{align*} {\mathscr L}_\phi^{\mathsf a} [w] := \left[ \frac{\partial}{\partial t} - \mathsf{a}(x,t) - \Delta_\phi \right] w (x,t) = \mathscr G(t, x, w(x,t)), \qquad t>0, \end{align*} on smooth metric measure spaces where the metric and potential are time dependent and evolve under a $({\mathsf k}, m)$-super Perelman-Ricci flow. A number of consequences, most notably, a parabolic Harnack inequality, a class of Hamilton type global curvature-free estimates and a general Liouville type theorem together with some consequences are established. Some special cases are presented to illustrate the strength of the results

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-\'Emery 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