A Lower Bound for the Eigenvalues of the Elliptic Dirichlet Problem for a General Domain in Terms of Its Characteristic Dimension
Shlomo Breuer, Joseph J. Roseman · SIAM Journal on Mathematical Analysis · 1978
We consider the Dirichlet eigenvalue problem on a general bounded domain $\mathcal{D} \subset R^n $, with a sufficiently regular boundary, for the equation $Lv - \lambda rv = 0$, where L is a linear, uniformly elliptic, selfadjoint differential operator of order $2s$, and r is a positive piecewise continuous function on $\mathcal{D}$. Quantities h, p, and $\delta $ are defined, where h is the supremum of the radii of all spheres contained in $\mathcal{D}$, p is an integer which characterizes its geometry, and $\delta = p^{{1 / {(2s)}}} h$; $\delta $ is called the characteristic dimension of $\mathcal{D}$ and is, in general, independent of the overall size of $\mathcal{D}$. It can then be shown that $\lambda _1 \geqq M({\kappa / K})\delta ^{ - 2s} $, where $\lambda _1 $ is the smallest eigenvalue, $\kappa $ is a parameter depending upon L, K is an upper bound for r and M depends only upon s and n. A detailed proof is given for $L = - abla ^2 $ and $r = 1$ in two dimensions and the extension to the general case is straightforward.