Local Gradient Estimates Research Articles

Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds

Let M be a complete noncompact Riemannian manifold of dimension n. In this paper, we derive a local gradient estimate for positive solutions of fast diffusion equations ∂ t u = Δu α , 1 ― 2 n < α < 1 n on M x (―∞, 0]. We also obtain a theorem of Liouville type for positive solutions of the fast diffusion equation.

Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote \({\Delta_g=-{\rm div}_g\nabla}\) the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation $$\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}$$ on (M, g). Here, p, q ≥ 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of $$u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)}$$ on (M, g) with initial data u(x, 0) > 0.

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

We establish local gradient estimates to solutions of general conformally invariant fully nonlinear second order elliptic equations. To cite this article: Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula

Dans la première partie de cet article, nous établissons des estimées locales de gradient pour les fonctions p-harmoniques à l’intérieur et au bord, sur les variétés riemanniennes générales. Grâce à ces estimations et suivant une idée récente de R. Moser, nous obtenons un théorème d’existence de solutions faibles au sens de la formulation d’ensemble de niveau pour le flot 1/H (inverse de la courbure moyenne) des hypersurfaces dans les variétés ambiantes ayant la propriété de la croissance optimale du volume. Dans la deuxième partie, nous considérons deux types d’équations paraboliques pour les fonctions p-harmoniques et nous établissons une estimation optimale du type de Li-Yau pour les solutions positives pour ces équations sur les variétés à courbure de Ricci non-négative. Nous montrons aussi une formule de monotonie des entropies associées à ces équations. Cette formule généralise un résultat antérieur du deuxième auteur pour l’équation de la chaleur linéaire. Comme application, nous montrons que toute variété riemannienne complète à courbure de Ricci positive ou nulle et admettant une inégalité logarithmique L p optimale est isométrique à l’espace euclidien.

Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds

In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation $$u_t=\Delta u+au\, {\rm log}\, u+bu\quad {\rm in}\,M$$ , where a, b are real constants, M is a complete noncompact Riemannian manifold. As a corollary, we give a local gradient estimate for the corresponding elliptic equation: $$\Delta u+au\,{\rm log}\, u+bu=0\quad {\rm in}\,M$$ , which improves and extends the result of Ma (J Funct Anal 241:374–382, 2006) and get a bound for the positive solution to this elliptic equation.

Autonomous object planar movement control in investigating scalar fields

A gradient autonomous object movement control algorithm is proposed. The algorithm ensures search and tracing of a scalar field isoline for a vehicle with one sensor. A substantial limitation is the possibility of obtaining only local gradient estimates at discrete times. The algorithm is based on input of an additional test signal to the control system. This enables the vehicle with one sensor to calculate the total gradient value.

Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

In this work we derive local gradient and Laplacian estimates of the Aronson–Bénilan and Li–Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let ( M , g ) (M,g) be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation \[ ∂ u ∂ t = Δ u + a u log ⁡ u + b u \frac {\partial u}{\partial t}=\Delta u+au\log u+bu \] on M × [ 0 , + ∞ ) M\times [0,+\infty ) , where a a , b b are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).

C 1, α regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C 1, α regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE.

On derivative estimation and the solution of least squares problems

Surface interpolation finds application in many aspects of science and technology. Two specific areas of interest are surface reconstruction techniques for plant architecture and approximating cell face fluxes in the finite volume discretisation strategy for solving partial differential equations numerically. An important requirement of both applications is accurate local gradient estimation. In surface reconstruction this gradient information is used to increase the accuracy of the local interpolant, while in the finite volume framework accurate gradient information is essential to ensure second order spatial accuracy of the discretisation. In this work two different least squares strategies for approximating these local gradients are investigated and the errors associated with each analysed. It is shown that although the two strategies appear different, they produce the same least squares error. Some carefully chosen case studies are used to elucidate this finding.

Properties of positive solutions to an elliptic equation with negative exponent

In this paper, we study some quantitative properties of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics as well as differential geometry. We can get some useful local gradient estimate and L 1 lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.

Degenerate Conformally Invariant Fully Nonlinear Elliptic Equations

We establish Liouville theorems for $$C_{loc}^{1,1}$$ , entire solutions and locally Lipschitz entire weak solutions to general degenerate conformally invariant fully nonlinear elliptic equations of second order. For applications to local gradient estimates of solutions of general conformally invariant fully nonlinear elliptic equations of second order, see [20].

Local gradient estimates for nonlinear elliptic equations

In this paper we obtain local L q , q ⩾ p , gradient estimates for weak solutions of elliptic equations of p-Laplacian type with small BMO coefficients.

LOCAL GRADIENT ESTIMATES FOR QUOTIENT EQUATIONS IN CONFORMAL GEOMETRY

In this paper, we extend the local gradient estimates established in [8, 10] to the conformal quotient equations. An existence of solutions of the conformal quotient equations follows from the local gradient estimates and the paper [14] of Gursky and Viaclovsky.

Conformal deformations of the smallest eigenvalue of the Ricci tensor

We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor is a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a Pucci type equation with respect to the Schouten tensor. We establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations. Combining it with the theory of fully nonlinear equations, we establish the existence of solutions for this equation.

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

We establish local gradient estimates to solutions of general conformally invariant fully nonlinear second order elliptic equations. To cite this article: Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds

In this paper, we study the local gradient estimate for the positive solution to the following equation: Δ u + a u log u + b u = 0 in M , where a < 0 , b are real constants, M is a complete non-compact Riemannian manifold. Our result is optimal in the sense when ( M , g ) is a complete non-compact expanding gradient Ricci soliton. By definition, ( M , g ) is called an expanding gradient Ricci soliton if for some constant c < 0 , it satisfies that Rc = c g + D 2 f , where Rc is the Ricci curvature, and D 2 f is the Hessian of the potential function f on M. We show that for a complete non-compact Riemannian manifold ( M , g ) , the local gradient bound of the function f = log u , where u is a positive solution to the equation above, is well controlled by some constants and the lower bound of the Ricci curvature.

Some gradient estimates for the heat equation on domains and for an equation by Perelman

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.

Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations

Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations

A Lioville Type Theorem for Minimizing Maps

Here we establish a Liouville type theorem for minimizing maps from R2 (or in general, from Rm) into a compact Riemannian manifold N. As a consequence of this, we prove a local gradient estimate for minimal solutions to a variational problem arise from planar ferromagnetism and anti-ferromagnetism. The latter can be applied to study the asymptotic behavior of entire solutions.