# Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators

## Related Articles

- The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids. Nazarov, S. // Journal of Mathematical Sciences;Oct2012, Vol. 186 Issue 2, p247
We study the spectrum of the Dirichlet problem for the Laplace operator in a cylindrical waveguide with periodic family of small (of diameter Îµ > 0) voids. Based on the asymptotic analysis of eigenvalues of the problem in a singularly perturbed periodicity cell, we show that the waveguide...

- Three Solutions for Inequalities Dirichlet Problem Driven by p(x)-Laplacian-Like. Zhou Qing-Mei; Ge Bin // Abstract & Applied Analysis;2013, p1
A class of nonlinear elliptic problems driven by p(x)-Laplacian-like with a nonsmooth locally Lipschitz potential was considered. Applying the version of a nonsmooth three-critical-point theorem, existence of three solutions of the problem is proved.

- Two-Dimensional Berezin-Li-Yau Inequalities with a Correction Term. Kovařík, Hynek; Vugalter, Semjon; Weidl, Timo // Communications in Mathematical Physics;Apr2009, Vol. 287 Issue 3, p959
We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas, [11].

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The article investigates the Laplacian operator's eigenvalue. It provides information on the applicability of the Laplacian's eigenvalue in solving the elliptic operators' homogenized problems and cites the methods of computing the solutions' convergence rate. It entails the variational...

- Asymptotic behavior of eigenvalues of the Laplace operator in thin infinite tubes. Grushin, V. V. // Mathematical Notes;Jun2009, Vol. 85 Issue 5/6, p661
In this paper, we obtain an asymptotic expansion for the eigenvalues of the Laplace operator with zero Dirichlet conditions in tubes, i.e., in infinite bent cylinders with internal torsion under uniform contraction of their cross-sections, with respect to a small parameter characterizing the...

- Three solutions to inequalities of Dirichlet problem driven by p( x)-Laplacian. Ge, Bin; Xue, Xiao-ping; Guo, Meng-shu // Applied Mathematics & Mechanics;Oct2010, Vol. 31 Issue 10, p1283
class of nonlinear elliptic problems driven by p( x)-Laplacian with a nonsmooth locally Lipschitz potential is considered. By applying the version of the nonsmooth three-critical-point theorem, the existence of three solutions to the problems is proved.

- Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions. Henrot, Antoine; Oudet, Edouard // Archive for Rational Mechanics & Analysis;Aug2003, Vol. 169 Issue 1, p73
In this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the ``stadium'', a convex hull of two identical tangent disks. We refute...

- Accuracy of the difference scheme of solving the eigenvalue problem for the Laplacian. Maiko, N.; Prikazchikov, V.; Ryabichev, V. // Cybernetics & Systems Analysis;Sep2011, Vol. 47 Issue 5, p783
The finite-difference approximation of the eigenvalue problem with the Dirichlet boundary conditions for the Laplacian in a two-dimensional domain of complex form is analyzed for accuracy and the error of eigenfunctions from the class $ W_2^2\left( \Omega \right) $ in the mesh norm of $...

- Asymptotics of eigenvalues of the Dirichlet problem in a skewed â„-shaped waveguide. Nazarov, S. // Computational Mathematics & Mathematical Physics;May2014, Vol. 54 Issue 5, p811
Asymptotics are constructed and justified for the eigenvalues of the Dirichlet problem for the Laplacian in a waveguide consisting of a unit strip and a semi-infinite strip joined at a small angle É› âˆˆ (0, Ï€/2). Some properties of the discrete spectrum are established, and open...