Abstract
We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds(M,g,e-ϕdv). We establish a Yang-type inequality of this problem. We also get universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below and any complete manifolds admitting eigenmaps to a sphere.
Highlights
Let (M, ⟨, ⟩) be an n-dimensional bounded compact Riemannian manifold, φ ∈ C2(M), and dμ = e−φdV, where dV is the Riemannian volume measure on (M, ⟨, ⟩)
We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds (M, g, e−φdV)
Many mathematicians have paid their attention to the eigenvalue problem of the drifting Laplacian on Riemannian manifolds
Summary
Many mathematicians have paid their attention to the eigenvalue problem of the drifting Laplacian on Riemannian manifolds (see [1,2,3]). They have studied the following eigenvalue problem:.
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