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1.
The eigenvalues and eigenfunctions of p-Laplacian on Finsler manifolds are defined to be critical values and critical points of its canonical energy functional. Based on it, we generalize some eigenvalue comparison theorems of p-Laplacian on Riemannian manifolds, such as Lichnerowicz type estimate, Obata type theorem and Mckean type theorem, to the Finsler setting. Not only that, the Lichnerowicz type estimate we obtained is even better than the corresponding one in Riemannian geometry.  相似文献   

2.
We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.  相似文献   

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4.
We consider the eigensolver named FEAST that was introduced by Polizzi in 2009 [1]. This solver, tailored to the (partial) solution of large sparse (generalized) eigenproblems offers good potential for parallelism and showed good robustness in our experiments. We briefly introduce the algorithm, point out some problems and give two examples. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

5.
A modified projection method for eigenvalues and eigenvectors of a compact operator T on a Banach space is defined and analyzed. The method is derived from the Kantorovich regularization for second-kind equations involving the operator T. It is shown that when T is a positive self-adjoint operator on a Hilbert space and the projections are orthogonal, the modified method always gives eigenvalue approximations which are at least as accurate as those obtained from the projection method. For self-adjoint operators, the required computation is essentially the same for both methods. Numerical computations for two integral operators are presented. One has T positive self-adjoint, while in the other T is not self-adjoint. In both cases the eigenvalue approximations from the modified method are more accurate than those from the projection method.  相似文献   

6.
Let Δ p denote the p-Laplacian operator and Ω be a bounded domain in . We consider the eigenvalue problem
for a potential V and a weight function m that may change sign and be unbounded. Therefore the functional to be minimized is indefinite and may be unbounded from below. The main feature here is the introduction of a value α(V, m) that guarantees the boundedness of the energy over the weighted sphere . We show that the above equation has a principal eigenvalue if and only if either m ≥ 0 and α(V, m) > 0 or m changes sign and α(V, m) ≥ 0. The existence of further eigenvalues is also treated here, mainly a second eigenvalue (to the right) and their dependence with respect to V and m.   相似文献   

7.
This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C~(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.  相似文献   

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9.
本文我们考虑一个带间断系数的特征值问题,使用Fourier G a lerk in方法求解.数值试验表明特征值收敛速度达到三阶,而特征函数收敛速度达到2.5阶.表明此方法对间断系数问题非常有效.  相似文献   

10.
Urs Miller  Lothar Gaul 《PAMM》2011,11(1):331-332
For the direct solution of quadratic eigenvalue problems of the form (λ2 M + P + Q ) x = 0 , a generalization of the Rayleigh quotient iteration is presented. Numerical simulations show good convergence for problems where the eigenvalues have nonzero imaginary part. The method is used to calculate eigenvalue paths of parameter dependent problems in structural dynamics. Bifurcations with double eigenvalues, which can occur in the path, are passed by using a perturbation of the velocity dependent matrix P . (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

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