Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces (original) (raw)
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Heliyon
In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equation Δ + () + () = 0, on complete smooth metric measure spaces (, , −) with ∞-Bakry-Émery Ricci tensor bounded from below, where is an arbitrary real constant, () and () are smooth functions. As an application, Liouvilletype theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on (, , −) with nonpositive weighted scalar curvature.
Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation
Journal of Mathematical Analysis and Applications
Let (M N , g, e −f dv) be a complete smooth metric measure space with ∞-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation Δ f − ∂ ∂t u(x, t) + q(x, t)u α (x, t) = 0, where (x, t) ∈ M N × (−∞, ∞) and α is an arbitrary constant. As Applications we prove a Liouville-type theorem for positive ancient solutions and Harnack-type inequalities for positive bounded solutions.
Harnack inequality for nonlinear parabolic equations under integral Ricci curvature bounds
arXiv (Cornell University), 2022
Let (M n , g) be a complete Riemannian manifold. In this paper, we establish a space-time gradient estimates for positive solutions of nonlinear parabolic equations ∂tu(x, t) = ∆u(x, t) + au(x, t)(log u(x, t)) b + q(x, t)A(u(x, t)), on geodesic balls B(O, r) in M with 0 < r ≤ r for p > n 2 when integral Ricci curvature k(p, 1) is small enough. By integrating the gradient estimates, we find the corresponding Harnack inequalities.
Gradient Estimates for Heat-Type Equations on Manifolds Evolving by the Ricci Flow
International Journal of Pure and Apllied Mathematics, 2014
In this paper, certain localized and global gradient estimates for all positive solutions to the geometric heat equation coupled to the Ricci flow either forward or backward in time are proved. As a by product, we obtain various Li-Yau type differential Harnack estimates. We also discuss the case when the diffusion operator is perturbed with the curvature operator (precisely, when the Laplacian is replaced with "∆−R(x, t)", R being the scalar operator). This is well generalised to the case of an adjoint heat equation under the Ricci flow.
Gradient estimates for heat kernels and harmonic functions
Journal of Functional Analysis, 2019
Let (X, d, µ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a "carré du champ". Assume that (X, d, µ, E) supports a scale-invariant L 2-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2, ∞]: (i) (G p): L p-estimate for the gradient of the associated heat semigroup; (ii) (RH p): L p-reverse Hölder inequality for the gradients of harmonic functions; (iii) (R p): L p-boundedness of the Riesz transform (p < ∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p ∈ (2, ∞), (i), (ii) (iii) are equivalent, while for p = ∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L 2-Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p = ∞, while for p ∈ (2, ∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.
Differential Harnack and logarithmic Sobolev inequalities along Ricci-harmonic map flow
Pacific Journal of Mathematics, 2015
This paper introduces a new family of entropy functionals which is proved to be monotonically nondecreasing along the Ricci-harmonic map heat flow. Some of the consequences of the monotonicity are combined to derive gradient estimates and Harnack inequalities for all positive solutions to the associated conjugate heat equation. We relate the entropy monotonicity and the ultracontractivity property of the heat semigroup, and as a result we obtain the equivalence of logarithmic Sobolev inequalities, conjugate heat kernel upper bounds and uniform Sobolev inequalities under Ricci-harmonic map heat flow.
Inventiones mathematicae, 2013
This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X, d, m). Our main results are: • A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X, d). • The equivalence of the heat flow in L 2 (X, m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional Ent m in the space of probability measures P(X). • The proof of density in energy of Lipschitz functions in the Sobolev space W 1,2 (X, d, m). • A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39, 40] and require neither the doubling property nor the validity of the local Poincaré inequality.
Archiv der Mathematik, 2015
The purpose of this paper is to study gradient estimates of Hamilton-Souplet-Zhang type for the following general heat equation ut = ∆V u + au log u + bu on noncompact Riemannian manifolds. As its application, we show a Harnack inequality for the positive solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extension and improvement of the work of Souplet-Zhang ([11]), Ruan ([10]), Yi Li ([7]), Huang-Ma ([6]), and Wu ([12]).
Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case
Springer INdAM Series, 2013
We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L 2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.