Let \(\varOmega \) be a bounded open domain in \(\mathbb {R}^n\) and \(\triangle _{\mu }=\sum _{j=1}^{n}\partial _{x_{j}}(\mu _{j}^2(x)\partial _{x_{j}})\) be a class of degenerate elliptic operator with continuous nonnegative coefficients \(\mu _{1},\mu _{2},\ldots ,\mu _{n}\). Denote by \(\lambda _{k}\) the kth Dirichlet eigenvalue of the self-adjoint degenerate elliptic operator \(-\triangle _{\mu }\) on \(\varOmega \). If the coefficients \(\mu _{1},\mu _{2},\ldots ,\mu _{n}\) satisfy some general assumptions, we give an explicit lower bound estimate of \(\lambda _{k}\). Moreover, if the coefficients \(\mu _{1}\ldots ,\mu _{n}\) are homogeneous functions with respect to a group of dilation, then we obtain an explicit sharp lower bound estimate for \(\lambda _{k}\), which has a polynomially growth in k of the order related to the homogeneous dimension. Finally, we also establish an upper bound estimate of \(\lambda _{k}\) for general self-adjoint degenerate elliptic operator \(\triangle _{\mu }\).

假设\（\ varOmega \）是\（\ mathbb {R} ^ n \）和\（\ triangle _ {\ mu} = \ sum _ {j = 1} ^ {n} \ partial _中的有界开放域{x_ {j}}（\ mu _ {j} ^ 2（x）\ partial _ {x_ {j}}）\）是一类具有连续非负系数\（\ mu _ {1}的简并椭圆算子， \ mu _ {2}，\ ldots，\ mu _ {n} \）。用\（\ lambda _ {k} \）表示\（\ varOmega \）上的自伴随退化椭圆算子\（-\ triangle _ {{mu} ）的第k个Dirichlet特征值。如果系数\（\ mu _ {1}，\ mu _ {2}，\ ldots，\ mu _ {n} \）满足一些一般假设，我们将给出\（\ lambda _ {k } \）。此外，如果系数\（\ mu _ {1} \ ldots，\ mu _ {n} \）对于一组扩张是齐次函数，则我们可以获得\（\ lambda _的显式尖锐下界估计{k} \），它在k中具有多项式增长，其阶数与齐次维相关。最后，我们还为一般自伴生退化椭圆算子\（\ triangle _ {{mu}）建立了\（\ lambda _ {k} \）的上限估计。