Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons
Let \(\Omega\) be a bounded domain inside a complete Riemannian manifold \((M,g)\). Let \(L_\phi:=\Delta-(\nabla\phi,\nabla\cdot)\) be the weighted Laplacian or the drifting Laplacian, where \(\phi\in C^\infty(M)\). Impose boundary conditions defined by the boundary operator \(\mathcal{B}u=\{u\oplus\partial_\nu u\oplus\dots\oplus\partial_\nu^{m-1}\}|_{\partial M}\), where \(\nu\) is the inward unit normal vector field. The operator \((-L_\phi)^m\) is self-adjoint with these boundary conditions with respect to the inner-product defined by the conformally equivalent measure \(e^{-\phi}\text{dvol}_g\). The authors consider the eigenvalue problem \((-L_\phi)^mu=\Lambda u\) in \(\Omega\) with these boundary conditions and obtain a general inequality for lower-order eigenvalues. They obtain some universal inequalities for lower-order eigenvalues in the case when the ambient space is Euclidean space or the unit sphere. They also obtain universal inequalities in the Gaussian and cylinder shrinking solitons. The first section is an introduction to the problem, the second section treats some preliminaries, and the third section establishes the main results.
scientific article; zbMATH DE number 6488356
scientific article