Domain-Independent Upper Bounds for Eigenvalues of Elliptic Operators
Domain-Independent Upper Bounds for Eigenvalues of Elliptic Operators
Let $\Omega \subseteq {\mathbb {R}^m}$ be a bounded open set, $\partial \Omega$ its boundary and $\Delta$ the Laplacian on ${\mathbb {R}^m}$. Consider the elliptic differential equation: (1) \[ - \Delta u = \lambda u\quad {\text {in}}\;\Omega ;\qquad u = 0\quad {\text {on}}\;\partial \Omega .\] It is known that the eigenvalues, …