紧黎曼流形上椭圆型方程的逐点梯度估计
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摘要
本文主要讨论带有非负Bakry-Emery Ricci曲率的紧致黎曼流形上一类半线性椭圆型方程解的梯度估计,利用"P-函数技术"与极大值原理得到了解的最优逐点梯度估计.
In this paper, we prove a pointwise gradient bound for a quasiliear elliptic equation on compact Riemannian manifolds with nonnegative Bakry-Emery Ricci curvature. We use the "P-function technique" and Maximum Principle to obtain the optimal pointwise gradient estimation for solutions.
In this paper, we prove a pointwise gradient bound for a quasiliear elliptic equation on compact Riemannian manifolds with nonnegative Bakry-Emery Ricci curvature. We use the "P-function technique" and Maximum Principle to obtain the optimal pointwise gradient estimation for solutions.
引文
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[2]Lott J.Some geometric properties of the Bakry-émery-Ricci tensor[J].Commentarii Mathematici Helvetici,2003,78(4):865-883.
[3]Bakry D,émery M.Diffusions hypercontractives[J].Séminaire de probabilités de Strasbourg,1985,19(1):177-206.
[4]Lott J,Villani C.Ricci curvature for metric-measure spaces via optimal transport[J].Annals of mathematics,2009,169(3):903-991.
[5]Sturm K T.On the geometry of metric measure spacesI[J].Acta mathematica,2006,196(1):65-131.
[6]Sturm K T.On the geometry of metric measure spaces.II[J].Acta mathematica,2006,196(1):133-177.
[7]Farina A,Valdinoci E.A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature[J].Discrete Contin.Dyn.Syst,2011,30(4):1139-1144.
[8]Caffarelli L,Garofalo N,Segala F.A gradient bound for entire solutions of quasi‐linear equations and its consequences[J].Communications on Pure and Applied Mathematics,1994,47(11):1457-1473.
[9]Payne L E.Some remarks on maximum principles[J].Journal d'Analyse Mathématique,1976,30(1):421-433.
[10]Sperb R P,Sperb R P.Maximum principles and their applications[M].New York:Academic Press,1981.
[11]Farina A,Sire Y,Valdinoci E.Stable solutions of elliptic equations on Riemannian manifolds[J].Journal of Geometric Analysis,2013,23(3):1157-1172.
[12]GILBARG D A,Trudinger N S.Elliptic partial differential equations of second order[M].New York:Springer,2001.