Eigenvalue Estimates For A Class Of Elliptic Operators In Divergence Form

In this paper,we mainly study two kinds of eigenvalue problems for a class of elliptic operator in divergence form:the first one is the clamped plate problem of the operator Lr;the second one is the buckling problem of the operator ?(?).For the clamped plate problem of the operator Lr,we will investigate upper bounds of eigenvalues on a compact self-shrinker.Using a family of appropriate trial functions,we get estimates for upper bounds of eigenvalues.For the buckling problem of the operator ?(?),we will study upper bounds of eigenvalues on a metric measure space(M,g,e-(?)dv),which satisfies Ric(?)?0.According to the properties of the metric measure space(M,g,e-(?)dv),we construct a family of trial functions.By making use of the trial functions,we derive estimates for upper bounds of higher eigenvalues.