The eigenvalue problem for −Δu=λu with Dirichlet boundary conditions for two‐dimensional regions. II. T‐shaped, cross‐shaped, and H‐shaped regions

1988 ◽  
Vol 83 (1) ◽  
pp. 133-144 ◽  
Author(s):  
Kazuo Hashimoto
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional

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>

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