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In this paper, we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in R n + 1 with speed ψ σ k α (λ) , where α > 1 k is a constant, σ k (λ) is the k-th elementary symmetric polynomial of the principal radii of curvature and ψ is a preassigned positive smooth function defined on S n . We prove that under some assumptions of ψ , the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known L p Christoffel-Minkowski problem u 1 - p (x) σ k (∇ 2 u + u I) = c ψ (x) for 1 < p < k + 1 . [ABSTRACT FROM AUTHOR]

Let (N, g) be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations Δ u (x) + a (x) u (x) log u (x) + b (x) u (x) = 0 on N where a(x) is C2-smooth while b(x) is C1 and its parabolic counterparts (Δ − ∂ ∂ t ) u (x , t) + a (x , t) u (x , t) log u (x , t) + b (x , t) u (x , t) = 0 on N × [0, ∞) where a(x, t) and b(x, t) are C2 with respect to x ∊ N while are C1 with respect to the time t. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a (k , m) -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results. [ABSTRACT FROM AUTHOR]

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has ‖ P t μ − P t ν ‖ 1 ≤ W 1 (μ , ν) t for all probability measures μ , ν where P t is the heat semigroup and W 1 is the ℓ 1 -Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the eigenvalue estimate λ 1 ≥ log ⁡ (2) / diam 2. [ABSTRACT FROM AUTHOR]

In this paper, we consider gradient estimates for positive solutions to the following equation △V u + aup log u = 0 on complete noncompact Riemannian manifold with k -dimensional Bakry-Émery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of |∇u| with respect to the lower bound of the Bakry-Émery Ricci curvature. [ABSTRACT FROM AUTHOR]

By using the perpetual cutoff method, we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K, N). This generalizes a main result of F. Münch who considers the case of CD(K, ∞) curvature. Hence, we answer a question raised by Münch. For that purpose, we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded Laplacian Δ and perpetual cutoff semigroup Ptw in our setting. [ABSTRACT FROM AUTHOR]

Let (M n , g , e - f d v) be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation Δ f - ∂ ∂ t u (x , t) + q (x , t) u (x , t) + a u (x , t) (ln u (x , t)) α = 0 , where (x , t) ∈ M × (- ∞ , ∞) and a, α are arbitrary constants. Under the assumption that the ∞ -Bakry-Émery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton's type and Souplet–Zhang's type) gradient estimates to positive smooth solutions of this equation. [ABSTRACT FROM AUTHOR]

Let (Mn, g, e- fdv) be a smooth metric measure space of dimensional n. Suppose that v is a positive weighted p-eigenfunction associated to the eigenvalue λ 1, p on M, namelyefdiv (e- f ⊥ v p- 2⊥ v) =- λ 1, pvp- 1, in the distribution sense. We first give a local gradient estimate for v provided the m-dimensional Bakry-Émery curvature Ricfm bounded from below. Consequently, we show that when Ricfm≥; 0 then v is constant if v is of sublinear growth. At the same time, we prove a Harnack inequality for weighted p-harmonic functions. Moreover, we show global sharp gradient estimates for weighted p-eigenfunctions. Then we use these estimates to study geometric structures at infinity when the first eigenvalue λ 1, p is maximal. Our achievements generalize several results proved earlier by Li-Wang, Munteanu-Wang ( [11,12,17,18]). [ABSTRACT FROM AUTHOR]

In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. In particular, we shall give a new proof for the capillary problem with zero gravity. [ABSTRACT FROM AUTHOR]

Some qualitative properties of positive smooth solutions to a generalized nonlinear parabolic equation involving f -Laplacian (L f) (L f − q (x , t) − ∂ ∂ t) w (x , t) = G (w (x , t)) , are discussed on M f × (− ∞ , + ∞) , where M f is a complete smooth metric measure space (with or without boundary), the potential function q (x , t) is smooth at least C 1 in x and C 0 in t , and G (w (x , t)) is a nonlinear smooth sourcing term. Local and global type space only (elliptic type) gradient estimates are established for this equation under the condition that Bakry-Émery Ricci curvature tensor is bounded from below. As an exploitation of the gradient estimates so derived we obtain a parabolic Harnack inequality and some Liouville type theorems for bounded ancient and eternal solutions. The approach adopted in this paper provides a unified treatment of a large class of nonlinear source terms. To demonstrate this further, cases where G (w) = a w ρ , a ∈ R , ρ ∈ (− ∞ , 0 ] ∪ [ 1 , + ∞) , G (w) = a w | log ⁡ w | γ , a ∈ R , a ≠ 0 , γ > 1 and G (w) = a w (log ⁡ w) γ a ≠ 0 , γ ≥ 1 are considered as specific examples. We further show that all the obtained results also hold on weighted manifolds with compact boundary under some lower boundedness assumptions on mean curvature of the boundary and Bakry-Émery Ricci curvature tensor. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group . In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a inequality), whereas the operator itself does not satisfy any inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. [Copyright &y& Elsevier]

Abstract: Let be a complete noncompact manifold with Ricci curvature bounded below. In this note, we derive a uniform bound for the solutions to the nonlinear equation where is a real constant. Our method is based on the refined global gradient estimates for the corresponding evolution equation, which is due to Yau [S.T. Yau, Harnack inequality for non-self-adjoint evolution equations, Math. Res. Lett. 2 (1995) 387–399]. We partially generalize the result of Yang [Y.Y. Yang, Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc. 136 (2008) 4095–4102]. In the particular case of complete Riemannian manifolds with nonnegative curvature, we get a sharp upper bound for the positive solutions; this upper bound is independent of the dimension of the manifolds. [Copyright &y& Elsevier]

Abstract: For the reflected diffusion generated by on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of of the Poisson equation in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger''s isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided. [Copyright &y& Elsevier]

When the sectional curvature of the target manifold is negative, we establish a Schwarz lemma for f -harmonic maps, if the dimension of the domain and the target is large, the result improves Theorem 3 in Chen and Zhao (2017) for the case of V = ∇ f. When the sectional curvature of the target is nonpositive, we obtain a Liouville theorem for the general V -harmonic maps, as a consequence, any V -harmonic function u , satisfying | u (x) | = o (r (x) ) , on a complete Riemannian manifold with nonnegative Bakry–Emery–Ricci curvature is a constant. We also give some applications on gradient Ricci solitons and gradient solitons with potential which are solutions to Ricci-harmonic flow. [ABSTRACT FROM AUTHOR]

A class of semilinear heat flows with general nonlinear reaction terms is considered on complete Riemannian manifolds with Ricci curvature bounded from below. Two types of (space-time and space only) gradient estimates are established for positive solutions to the flow, and the corresponding Harnack inequalities are obtained to allow for comparison of solutions. Some specific examples of the reaction term such as logarithmic reaction, Fisher-KPP and Allen-Cahn equations are discussed as applications of the estimates so derived. Referring to logarithmic nonlinearities, some discussions are made on Liouville type properties of bounded solutions. [ABSTRACT FROM AUTHOR]