This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is...This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation： the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity （which was known years ago）, as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at presented at the end of Section 2 to infinity is also studied. Two examples are illustrate the value of the investigation.展开更多
By using diffusion process with absorbing boundary,some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+▽h on a non-compact complete Riemannian manifold. The resulting estimates contain Mc...By using diffusion process with absorbing boundary,some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+▽h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean’s estimate for ▽h=0.Moreover,the first Dirichlet eigen- value for elliptic operators on R^d and the first mixed eigenvalue are also studied.Some examples show that our estimates can be sharp even for ▽h≠0.展开更多
基金Acknowledgements The authors would like to thank Professors Yonghua Mao and Yutao Ma for their helpful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003）, the Specialized Research Fund for the Doctoral Program of Higher Education （Grant No. 20100003110005）, the ＇985＇ project from the Ministry of Education in China, and the Fundamental Research Funds for the Central Universities.
文摘This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation： the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity （which was known years ago）, as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at presented at the end of Section 2 to infinity is also studied. Two examples are illustrate the value of the investigation.
基金Research supported in part by NFSCthe State Education Commission of China
文摘By using diffusion process with absorbing boundary,some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+▽h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean’s estimate for ▽h=0.Moreover,the first Dirichlet eigen- value for elliptic operators on R^d and the first mixed eigenvalue are also studied.Some examples show that our estimates can be sharp even for ▽h≠0.
