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.     

Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds

We derive the gradient estimates and Harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations (Δ − ∂/∂t) u(x, t) + h(x, t)u^α(x, t) = 0 and Δu + b · u + hu^α = 0 on Riemannian manifolds. We also obtain a theorem of Liouville type for positive solutions of the nonlinear elliptic equation.     

Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

We investigate stability issues for Einstein-scalar field Lichnerowicz equations in the inhomogeneous context of a compact Riemannian manifold. We prove that stability holds true when the dimension n is such that n ≤ 5 and fails to hold in general when n ≥ 6.     

Gradient estimates for doubly nonlinear parabolic equations

Local gradient estimates for weak solutions of the equation

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$ on ${M \times [0, + \infty)}Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation

\frac?u?t = Dfu +au log u + bu \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu      7. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds      Yunyan Yang 《Proceedings of the American Mathematical Society》2008,136(11):4095-4102 Let be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation on , where , 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).       8. Partial Schauder estimates for second-order elliptic and parabolic equations      Hongjie Dong  Seick Kim 《Calculus of Variations and Partial Differential Equations》2011,40(3-4):481-500 We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H?lder semi-norms not with respect to all, but only with respect to some of the independent variables.      9. On the p-Laplacian Lichnerowicz equation on compact Riemannian manifolds      Chen  Nanbo  Liu  Xiaochun 《中国科学 数学(英文版)》2021,64(10):2249-2274 In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from the variational method, we obtain the non-existence, existence, and multiplicity results under some given assumptions.      10. Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds      Li Chen  Wenyi Chen 《Annals of Global Analysis and Geometry》2009,35(4):397-404 In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation , 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: , 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.       11. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations      Marie-Françoise Bidaut-Veron  Laurent Veron 《Inventiones Mathematicae》1991,106(1):489-539 We study the asymptotics and the global solutions of the following Emden equations: –u=e u in a 3-dim domain (>0) or –u=u q +|x|–2 u (q>1) in anN-dim domain. Precise behaviour is obtained by the use of Simon's results on analytic geometric functionals. In the case of the first equation, or the second equation with =0 andq=(N+1)/(N–3) (N>3), we point out how the asymptotics are described via the Moebius group onS N–1. For a conformally invariant equation –u=|u|4/(N–2) u+|x|–2 u(=±1) we prove the existence of a new type of solution of the formu(x)=|x|(2–N)/2((Ln|x|)(x/|x|)) where is defined onS N–1 and C (;O(N)). Finnally, we extend and simplify the results of Gidas and Spruck on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Licherowicz-Weitzenböck formula.      12. Eigenvalue estimates for a class of elliptic differential operators on compact manifolds      Hilário Alencar  Gregório Silva Neto  Detang Zhou 《Bulletin of the Brazilian Mathematical Society》2015,46(3):491-514       13. Gradient estimates for higher order elliptic equations on nonsmooth domains      Sun-Sig Byun  Seungjin Ryu 《Journal of Differential Equations》2011,250(1):243-263 We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical Wm,p, m=1,2,…, 1<p<∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.      14. Global weighted estimates for second-order nondivergence elliptic and parabolic equations      Fengping Yao 《Proceedings Mathematical Sciences》2018,128(1):11 In this paper, we obtain the global weighted \(L^p\) estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary, we obtain \(L^p\)-type regularity estimates for such equations.      15. Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds      Li Ma 《Journal of Functional Analysis》2006,241(1):374-382 In this paper, we study the local gradient estimate for the positive solution to the following equation:       16. Gradient estimates of Poisson equations on Riemannian manifolds and applications      Liming Wu   《Journal of Functional Analysis》2009,257(12):330-4033 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 supxM|G|(x) 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.      17. Gradient estimates in Orlicz space for nonlinear elliptic equations      Sun-Sig Byun  Fengping Yao  Shulin Zhou 《Journal of Functional Analysis》2008,255(8):1851-1873 In this paper we generalize gradient estimates in Lp space to Orlicz space for weak solutions of elliptic equations of p-Laplacian type with small BMO coefficients in δ-Reifenberg flat domains. Our results improve the known results for such equations using a harmonic analysis-free technique.      18. 紧流形上椭圆微分算子预解式的一致L^p-L^q估计——献给陆善镇教授75华诞          下载免费PDF全文 王华  尧小华 《中国科学:数学》2014,44(5):571-586 假设n和m是两个正整数,P（x,D）是定义在维数为n的紧致无边流形M上的一般m阶椭圆自伴微分算子.在一定条件下,本文主要证明微分算子P（x,D）的预解式的一致L^p-L^q估计,其中n〉m≥2,（p,q）在Sobolev线上并满足1/p-1/q=m/n,p≤2（n＋1）/n＋3,q≥2（n＋1）/n-1.本文的一个核心引理是建立曲面Σx={ξ∈Tx^＊（M）：p（x,ξ）=1}上测度的Fourier变换衰减估计的具体表达式,并利用它来得到局部算子的一致L^p-L^q估计.      19. Heat flow method for Lichnerowicz type equations on closed manifolds      Li?MaEmail author  Yuhua?Sun  Xiao?Tang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,79(2):261-270 In this paper, we establish existence results for positive solutions to the Lichnerowicz equations of the following type in closed manifolds -Du = A(x)u-p - B(x)uq,    in M,-\Delta u = A(x)u^{-p} - B(x)u^{q},\quad {{\rm in}}\, M,      20. Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds      Guangyue Huang  Zhijie Huang  Haizhong Li 《Annals of Global Analysis and Geometry》2013,43(3):209-232 Let (M n , g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: $$u_t=\Delta u+au\log u+bu$$ on M n  × [0,T], where a, b are two real constants. We derive local gradient estimates of the Li-Yau type for positive solutions of the above equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results extend the ones of Davies in Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol 92, Cambridge University Press, Cambridge,1989, and Li and Xu in Adv Math 226:4456–4491 (2011).

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Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation

on , where , 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).

     

Partial Schauder estimates for second-order elliptic and parabolic equations

We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H?lder semi-norms not with respect to all, but only with respect to some of the independent variables.     

On the p-Laplacian Lichnerowicz equation on compact Riemannian manifolds

In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from the variational method, we obtain the non-existence, existence, and multiplicity results under some given assumptions.     

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

Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations

We study the asymptotics and the global solutions of the following Emden equations: –u=e ^u in a 3-dim domain (>0) or –u=u ^q +|x|^–2 u (q>1) in anN-dim domain. Precise behaviour is obtained by the use of Simon's results on analytic geometric functionals. In the case of the first equation, or the second equation with =0 andq=(N+1)/(N–3) (N>3), we point out how the asymptotics are described via the Moebius group onS ^N–1. For a conformally invariant equation –u=|u|^4/(N–2) u+|x|^–2 u(=±1) we prove the existence of a new type of solution of the formu(x)=|x|(^2–N)/2((Ln|x|)(x/|x|)) where is defined onS ^N–1 and C (;O(N)). Finnally, we extend and simplify the results of Gidas and Spruck on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Licherowicz-Weitzenböck formula.     

Gradient estimates for higher order elliptic equations on nonsmooth domains

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical Wm,p, m=1,2,…, 1<p<∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.     

Global weighted estimates for second-order nondivergence elliptic and parabolic equations

In this paper, we obtain the global weighted \(L^p\) estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary, we obtain \(L^p\)-type regularity estimates for such equations.     

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:

     

Gradient estimates of Poisson equations on Riemannian manifolds and applications

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 sup_xM|G|(x) 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.     

Gradient estimates in Orlicz space for nonlinear elliptic equations

In this paper we generalize gradient estimates in Lp space to Orlicz space for weak solutions of elliptic equations of p-Laplacian type with small BMO coefficients in δ-Reifenberg flat domains. Our results improve the known results for such equations using a harmonic analysis-free technique.     

紧流形上椭圆微分算子预解式的一致L^p-L^q估计——献给陆善镇教授75华诞

假设n和m是两个正整数,P（x,D）是定义在维数为n的紧致无边流形M上的一般m阶椭圆自伴微分算子.在一定条件下,本文主要证明微分算子P（x,D）的预解式的一致L^p-L^q估计,其中n〉m≥2,（p,q）在Sobolev线上并满足1/p-1/q=m/n,p≤2（n＋1）/n＋3,q≥2（n＋1）/n-1.本文的一个核心引理是建立曲面Σx={ξ∈Tx^＊（M）：p（x,ξ）=1}上测度的Fourier变换衰减估计的具体表达式,并利用它来得到局部算子的一致L^p-L^q估计.     

Heat flow method for Lichnerowicz type equations on closed manifolds

In this paper, we establish existence results for positive solutions to the Lichnerowicz equations of the following type in closed manifolds

Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds

Let (M ⁿ , g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: $$u_t=\Delta u+au\log u+bu$$ on M ⁿ  × [0,T], where a, b are two real constants. We derive local gradient estimates of the Li-Yau type for positive solutions of the above equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results extend the ones of Davies in Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol 92, Cambridge University Press, Cambridge,1989, and Li and Xu in Adv Math 226:4456–4491 (2011).