Solution of a Nonlinear Eigenvalue Problem Using Signed Singular Values

2017 ◽  
Vol 7 (4) ◽  
pp. 799-809
Author(s):  
Kouhei Ooi ◽  
Yoshinori Mizuno ◽  
Tomohiro Sogabe ◽  
Yusaku Yamamoto ◽  
Shao-Liang Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Computational Efficiency ◽  
Numerical Experiments ◽  
Nonlinear Eigenvalue Problem ◽  
Singular Values ◽  
Singular Value ◽  
Scaling Exponent ◽  
Strong Nonlinearity ◽  
Constant Matrix ◽  
Nonlinear Eigenvalue

AbstractWe propose a robust numerical algorithm for solving the nonlinear eigenvalue problem A(ƛ)x = 0. Our algorithm is based on the idea of finding the value of ƛ for which A(ƛ) is singular by computing the smallest eigenvalue or singular value of A(ƛ) viewed as a constant matrix. To further enhance computational efficiency, we introduce and use the concept of signed singular value. Our method is applicable when A(ƛ) is large and nonsymmetric and has strong nonlinearity. Numerical experiments on a nonlinear eigenvalue problem arising in the computation of scaling exponent in turbulent flow show robustness and effectiveness of our method.

scholarly journals Lower bounds for Orlicz eigenvalues

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ariel Salort
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Dirichlet Boundary Conditions ◽  
Young Functions ◽  
New Strategies ◽  
Nonlinear Eigenvalue

<p style='text-indent:20px;'>In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.</p><p style='text-indent:20px;'>We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.</p>

Sign in / Sign up

Export Citation Format

Share Document