The lower bounds of the first eigenvalues for the biharmonic operator on manifolds

Abstract

In this paper, we use the Reilly formula and the Hessian comparison theorem to estimate the lower bounds of the first eigenvalues for the biharmonic operator eigenvalue problems (buckling problem and clamped plate problem) on manifolds. MSC: 35P15; 53C20

Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

“…For applications, following closely the idea of Hessian comparison estimates applied in [11,19,25] on different problems, we discuss lower bound estimates for the first eigenvalues on the weighted geodesic ball whose radius does not exceed the injectivity radius and submanifolds having bounded weighted mean curvature. In fact, our estimates extend the results of [11,25,38] when the supremum of the sectional curvature inside the ball is nonnegative.…”

“…3), we establish a weighted form of Escobar-Lichnerowicz-Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian defining buckling problem on weighted manifolds with generalized Ricci curvature bounded from below by a nonnegative constant. This result is a generalization of [17,38] in the case of bounded domains with smooth boundary in Riemannian manifolds. See also similar results [26,35,36,39] for the p-Laplacian or drifting Laplacian eigenvalue problems.…”

“…McKean's lower bound for weighted p-fundamental tone has been recently studied by the first author on a complete noncompact simply connected smooth metric measure space with sectional curvature bounded from above by negative constant [2,6]. Theorem 3.1 extends the result of Zhang and Zhao [38,Theorem 3.1], where they considered the first eigenvalue λ 1 (N) for buckling problem on an n-dimensional compact connected Riemannian manifold N with smooth boundary and Ric(N) ≥ K > 0 and obtained…”

“…First, we state an important lemma on Hessian comparison (see [16] for details), which will help realize the desired estimates. This idea has been applied previously by several authors [6,11,19,38] to different settings.…”

Boundary Value Problem of the Operator ⊕k Related to the Biharmonic Operator and the Diamond Operator

“…Finally, the heat flow in a nonuniform rod without sources accompanied with initial-boundary conditions [4]. These types of problems inevitably associate with the partial differential operators-for example, the Laplace operator [5,6], the ultrahyperbolic operator [7,8], and the biharmonic operator [9,10].…”