Yang-type inequalities for weighted eigenvalues of a second order uniformly elliptic operator with a nonnegative potential

Sun, He-Jun

doi:10.1090/S0002-9939-10-10321-9

Yang-type inequalities for weighted eigenvalues of a second order uniformly elliptic operator with a nonnegative potential
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Proc. Amer. Math. Soc. 138 (2010), 2827-2837 Request permission

Abstract:

In this paper, we investigate the Dirichlet weighted eigenvalue problem of a second order uniformly elliptic operator with a nonnegative potential on a bounded domain $\Omega \subset \mathbb {R}^n$. First, we prove a general inequality of eigenvalues for this problem. Then, by using this general inequality, we obtain Yang-type inequalities which give universal upper bounds for eigenvalues. An explicit estimate for the gaps of any two consecutive eigenvalues is also derived. Our results contain and extend the previous results for eigenvalues of the Laplacian, the Schrödinger operator and the second order elliptic operator on a bounded domain $\Omega \subset \mathbb {R}^n$.

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