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Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that $$\mu_1(g) \operatorname{...

I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$ $$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$ where the weights $a>0$ and $w\in L^{\infty}$ (and ...

Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$. We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator $$ -\Delta:\ H^...