For Riemannian manifolds with a measure,we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below,generalizing the classical ones of Yau(i.e.,when f is constant).
In this paper,we study the Kato's inequality on locally finite graphs.We also study the application of Kato's inequality to Ginzburg-Landau equations on such graphs.Interesting properties of elliptic and parabolic equations on the graphs and a Liouville type theorem are also derived.
In this article,the authors consider a class of semilinear elliptic equations on fractal sets under some new conditions,which are more weaker than those in usual cases. The authors get the non-trivial and non-negative solution of the zero boundary Dirichlet problem using Mountain Pass Lemma.
First, we review the authors’ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.
Let Ω be a connected bounded domain in Rn. Denote by λi the i-th eigenvalue of the Laplacian operator with any order p:{0(-△)pu =λu u =■u/→■n =···=■→■np-1 p-u1=0 in Ω,on ■Ω.In this article, we give some expressions for upper bound of the (k + 1)-th eigenvalue λk+1 in terms of the first k eigenvalues.
