Gradient Estimates For Positive Solutions Research Articles

Gradient estimates for Yamabe type equations under different curvature conditions and applications

The aim of this paper is to give various aspects of gradient estimates for positive solutions to the Yamabe type equation ut=Δfu+au+buα. We study the problem in many different settings. As result, we obtain several types of gradient estimates for the solution. When the manifold is complete without boundary, we study gradient estimates of Shi type, and of Hamilton type. Moreover, we also derive gradient estimates if the manifold is evolved under the (k,m)-super Perelman-Ricci flows. Besides, we prove gradient estimates under the integral Ricci curvature bounds. Finally, when the manifold has compact boundary, we investigate gradient estimates with Dirichlet boundary conditions. Our theorems generalize and improve many previous works.

Space-only gradient estimates of Schrödinger equation with Neumann boundary condition under integral Ricci curvature bounds

Assume that M (M⊆N) is an n-dimensional compact Riemannian submanifold with boundary, satisfying the integral Ricci curvature assumption:D2supx∈N⁡(∮B(x,d)|Ric−|pdy)1p<κ for κ=κ(n,p) small enough, p>n/2, where diam(M)≤D. The boundary of M needs to satisfy the interior rolling R-ball condition. We prove a Hamilton type gradient estimate for positive solutions to the parabolic Schrödinger equation with Neumann boundary conditions, on a compact Riemannian submanifold M with boundary ∂M, satisfying the integral Ricci curvature assumption.

Souplet–Zhang and Hamilton‐type gradient estimates for non‐linear elliptic equations on smooth metric measure spaces

AbstractIn this article, we present new gradient estimates for positive solutions to a class of non‐linear 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.

Gradient estimates for a nonlinear elliptic equation on a smooth metric measure space

Let (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type result for such equation in the case a &lt; 0 under the m-dimensions Bakry-Émery Ricci curvature.

Gradient estimates for a weighted parabolic equation under geometric flow

Let \((M^{n},g,e^{-\phi }dv)\) be a weighted Riemannian manifold evolving by geometric flow \(\frac{\partial g}{\partial t}=2\,h(t),\,\,\,\frac{\partial \phi }{\partial t}=\Delta \phi \). In this paper, we obtain a series of space-time gradient estimates for positive solutions of a parabolic partial equation $$\begin{aligned} (\Delta _{\phi }-\partial _{t})u(x,t)=q(x,t)u^{a+1}(x,t)+p(x,t)A(u(x,t))),\,\,\,\,(x,t)\in M\times [0,T], \end{aligned}$$on a weighted Riemannian manifold under geometric flow. By integrating the gradient estimates, we find the corresponding Harnack inequalities.

Geometric functionals for the p-Laplace operator on the graph

Let G=(V,E) be a connected finite graph. In the first part of this paper we assume that G satisfies the CDpψ condition for p>1 and some C1, concave function ψ:(0,+∞)→R. Based on the gradient estimate for positive solutions to the p-Laplace parabolic equation on G, we establish evolving formulas for the Fisher information, Shannon entropy and Perelman's Wp-functional along the p-Laplace parabolic equation on G. In the second part of this paper, we prove monotonicity of a p-parabolic frequency on G without any curvature condition, which generalizes Colding-Minicozzi II's monotone formula, both from Laplace operator to the p-Laplace operator, and also from the manifold to the graph setting. From the monotonicity, we get backward uniqueness for the p-Laplace operator on the graph.

Gradient estimates of a nonlinear Lichnerowicz equation under the Finsler-geometric flow

Let (Mn, F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow , where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we study gradient estimates for positive solutions of a nonlinear Lichnerowicz equation under Finsler-geometric flow , where c, p are two constants and p > 0. We obtain a global estimate and a Harnack estimate for positive solutions. Our results are also natural extension of similar results on Riemannian-geometric flow.

Aronson–Bénilan estimates for weighted porous medium equations under the geometric flow

In this paper, we study Aronson–Bénilan gradient estimates for positive solutions of weighted porous medium equations coupled with the geometric flow on a complete measure space . As an application, by integrating the gradient estimates, we derive the corresponding Harnack inequalities.

Quantitative Sobolev Extensions and the Neumann Heat Kernel for Integral Ricci Curvature Conditions

We prove the existence of Sobolev extension operators for certain uniform classes of domains in a Riemannian manifold with an explicit uniform bound on the norm depending only on the geometry near their boundaries. We use this quantitative estimate to obtain uniform Neumann heat kernel upper bounds and gradient estimates for positive solutions of the Neumann heat equation assuming integral Ricci curvature conditions and geometric conditions on the possibly non-convex boundary. Those estimates also imply quantitative lower bounds on the first Neumann eigenvalue of the considered domains.

The Rigidity and Stability of Gradient Estimates

In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with non-negative Gaussian curvature, the rigidity of the sharp Li-Yau gradient estimate for positive solutions to heat equations and the related estimates for Dirichlet Green’s functions on Riemannian manifolds with non-negative Ricci curvature. Moreover, we also obtain the corresponding stability results.

Gradient estimates for the weighted Lichnerowicz equation on smooth metric measure spaces

We consider the weighted Lichnerowicz equation △fu+cuσ=0 on smooth metric measure spaces, where c≥0,σ≤1 are real constants. A local gradient estimate for positive solutions to this equation is derived, and as an application we give a corresponding Harnack inequality.

Gradient estimates of a nonlinear elliptic equation for the $V$-Laplacian on noncompact Riemannian manifolds

In this paper, we consider gradient estimates for positive solutions to the following equation $$\triangle_V u+au^p\log u=0$$ on complete noncompact Riemannian manifold with $k$-dimensional Bakry-Emery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of $ \nabla u $ with respect to the lower bound of the Bakry-Emery Ricci curvature.

Gradient estimates for positive solutions of heat equations under Finsler-Ricci flow

We establish first order gradient estimates for positive solutions of the heat equations under closed Finsler-Ricci flow with weighted Ricci curvature bounded from below. As an application, we derive the corresponding Harnack inequality. Our results generalize the previous known related results.

Gradient estimates for a parabolic 𝑝-Laplace equation with logarithmic nonlinearity on Riemannian manifolds

In this paper, we study gradient estimates for a parabolic p p -Laplace equation with logarithmic nonlinearity, which is related to the L p L^p -log-Sobolev constant on Riemannian manifolds. We prove a global Li-Yau type gradient estimate and a Hamilton type gradient estimate for positive solutions to a parabolic p p -Laplace equation with logarithmic nonlinearity on compact Riemannian manifolds with nonnegative Ricci curvature. As applications, the corresponding Harnack inequalities are derived.

Sharp gradient estimates on weighted manifolds with compact boundary

&lt;p style='text-indent:20px;'&gt;In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.&lt;/p&gt;

Gradient estimates for a weighted nonlinear parabolic equation and applications

Abstract This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.

Upper Bounds on the First Eigenvalue for the p-Laplacian

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the p-Laplacian $$\begin{aligned} \Delta _{p}u=-\lambda |u|^{p-2}u \end{aligned}$$ with $$p>1$$ on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the p-Laplacian.

Gradient Estimates for p-Laplacian Lichnerowicz Equation on Noncompact Metric Measure Space

In this paper, the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation $${\Delta _{p,f}}u + c{u^\sigma } = 0$$ on noncompact smooth metric measure space, where c is a nonnegative constant, and p, σ (1 < p ≤ 2, σ ≤ p - 1) are real constants. Moreover, by the gradient estimate, they can get the corresponding Liouville theorem and Harnack inequality.

Gradient estimates of a nonlinear elliptic equation for the V-Laplacian

This paper studies gradient estimates for positive solutions of the nonlinear elliptic equation $$\begin{aligned} \Delta _V(u^p)+\lambda u=0,\quad p\ge 1, \end{aligned}$$on a Riemannian manifold (M, g) with k-Bakry–Emery Ricci curvature bounded from below. We consider both the case where M is a compact manifold with or without boundary and the case where M is a complete manifold.

Local parabolic and elliptic gradient estimates for a generalized heat-type equation under the Yamabe flow

In this paper, we investigate local parabolic (Li-Yau's type and J. Li's type) and local elliptic (Hamilton's type Shouplet-Zhang's type) gradient estimates for positive solutions to a generalized heat-type equation of (Δϕ−∂t)u=huq under the Yamabe flow. As applications of local parabolic gradient estimates, we establish associated Harnack inequalities.