Estimates of Dirichlet eigenvalues for degenerate $\triangle _{\mu }$-Laplace operator

Abstract

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 $\triangle _{\mu }$.