共查询到20条相似文献,搜索用时 0 毫秒
1.
Soojung Kim 《Journal of Differential Equations》2018,264(3):1613-1660
We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds, where an even function is supposed to be strictly convex on . Under the assumption that either or its convex conjugate with some structural condition, we establish a (locally) uniform ABP type estimate and the Krylov–Safonov type Harnack inequality on Riemannian manifolds with the use of an intrinsic geometric quantity to the operator. Here, the -regularities of F and account for degenerate and singular operators, respectively. 相似文献
2.
We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by −κ for κ≥0. In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such M, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved. 相似文献
3.
Yue WANG 《Frontiers of Mathematics in China》2010,5(4):727-746
We derive the gradient estimates and Harnack inequalities for positive solutions of the diffusion equation u
t
= Δu
m
on Riemannian manifolds. Then, we prove a Liouville type theorem. 相似文献
4.
B. A. Khudaigulyev 《Differential Equations》2012,48(2):255-263
We study the behavior of nonnegative solutions of the Dirichlet problem for a linear elliptic equation with a singular potential
in the ball B = B(0,R) ⊂ R
n
(n ≥ 3), R ≤ 1. We find an exact condition on the potential ensuring the existence or absence of a nonnegative solution of that problem. 相似文献
5.
In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)?−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M)?−k, which generalizes a result of L. Ni (2004, 2006) [20] and [21]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds. 相似文献
6.
7.
Li Jiayu 《Journal of Functional Analysis》1991,100(2)
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)uα(x, t) = 0 and Δu + b · u + huα = 0 on Riemannian manifolds. We also obtain a theorem of Liouville type for positive solutions of the nonlinear elliptic equation. 相似文献
8.
In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u q, q > 1, on spherically symmetric noncompact Riemannian manifolds. 相似文献
9.
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). 相似文献
10.
Shengbing Deng 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(3):859-881
Let (M,g) be a smooth compact Riemannian manifold of dimension n≥3. We are concerned with the following asymptotically critical elliptic problem
(0.1) 相似文献
11.
A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived. 相似文献
12.
Yong Hah Lee 《Mathematische Annalen》2000,318(1):181-204
We prove that if the s-harmonic boundary of a complete Riemannian manifold consists of finitely many points, then the set of bounded energy finite
solutions for certain nonlinear elliptic operators on the manifold is one to one corresponding to , where l is the cardinality of thes-harmonic boundary. We also prove that the finiteness of cardinality of s-harmonic boundary is a rough isometric invariant, moreover, in this case, the cardinality is preserved under rough isometries
between complete Riemannian manifolds. This result generalizes those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of
Kim and the present author, of Holopainen, and of the present author, but with different techniques which are demanded by
the peculiarity of nonlinearity.
Received October 13, 1999 / Revised November 23, 1999 / Published online July 20, 2000 相似文献
13.
Seick Kim Soojung Kim Ki-Ahm Lee 《Calculus of Variations and Partial Differential Equations》2014,49(1-2):669-706
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré’s result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature. 相似文献
14.
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:
15.
S. A. Korol’kov 《Siberian Mathematical Journal》2008,49(6):1051-1061
We study the problem of solvability of some boundary value problems on noncompact Riemannian manifolds with ends. We obtain the conditions for existence and uniqueness of solutions to the problems as well as the conditions for the fulfillment of Liouville-type theorems for harmonic functions on the manifolds. 相似文献
16.
17.
Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem
has a K-peaks solution, whose peaks collapse, as ε goes to zero, to an isolated local minimum point of the scalar curvature. Here p > 2 if N = 2 and .
E. N. Dancer was partially supported by the ARC. A. M. Micheletti and A. Pistoia are supported by Mi.U.R. Project “Metodi
variazionali e topologici nello studio di fenomeni non lineari”. 相似文献
18.
We study L-harmonic functions (solutions of the stationary Schrödinger equation) on arbitrary noncompact Riemannian manifolds with finitely many ends. We establish some existence and uniqueness results, and obtain sharp dimension estimates for L-harmonic functions on such manifolds. 相似文献
19.
We show that the number of solutions of a nonlinear elliptic problem on a Riemannian manifold depends on the topological properties of the manifold. In particular we consider the Lusternik-Schnirelmann category and the Poincaré polynomial of the manifold. 相似文献
20.
Paolo Antonini Dimitri Mugnai Patrizia Pucci 《Journal de Mathématiques Pures et Appliquées》2007,87(6):582-600
We prove maximum and comparison principles for weak distributional solutions of quasilinear, possibly singular or degenerate, elliptic differential inequalities in divergence form on complete Riemannian manifolds. A new definition of ellipticity for nonlinear operators on Riemannian manifolds is introduced, covering the standard important examples. As an application, uniqueness results for some related boundary value problems are presented. 相似文献