A Generic Property for the Eigenfunctions of the Laplacian
نویسندگان
چکیده
In this work we show that, generically in the set of C2 bounded regions of Rn, n ≥ 2, the inequality R Ω φ 3 6= 0 holds for any eigenfunction of the Laplacian with either Dirichlet or Neumann boundary conditions.
منابع مشابه
Lipschitz regularity of the eigenfunctions on optimal domains
We study the optimal sets Ω∗ ⊂ R for spectral functionals F ( λ1(Ω), . . . , λp(Ω) ) , which are bi-Lipschitz with respect to each of the eigenvalues λ1(Ω), . . . , λp(Ω) of the Dirichlet Laplacian on Ω, a prototype being the problem min { λ1(Ω) + · · ·+ λp(Ω) : Ω ⊂ R, |Ω| = 1 } . We prove the Lipschitz regularity of the eigenfunctions u1, . . . , up of the Dirichlet Laplacian on the optimal se...
متن کاملA new virtual leader-following consensus protocol to internal and string stability analysis of longitudinal platoon of vehicles with generic network topology under communication and parasitic delays
In this paper, a new virtual leader following consensus protocol is introduced to perform the internal and string stability analysis of longitudinal platoon of vehicles under generic network topology. In all previous studies on multi-agent systems with generic network topology, the control parameters are strictly dependent on eigenvalues of network matrices (adjacency or Laplacian). Since some ...
متن کاملOn Generalization of Sturm-Liouville Theory for Fractional Bessel Operator
In this paper, we give the spectral theory for eigenvalues and eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding eigenfunctions are orthogonal. In this paper, we give the spectral theory for eigenvalues and eigenfunctions...
متن کاملThe squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The wo...
متن کاملCoordinate finite type invariant surfaces in Sol spaces
In the present paper, we study surfaces invariant under the 1-parameter subgroup in Sol space $rm Sol_3$. Also, we characterize the surfaces in $rm Sol_3$ whose coordinate functions of an immersion of the surface are eigenfunctions of the Laplacian $Delta$ of the surface.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007