We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.

In this paper, we study, in some new ways, the estimation of unimodal densities. Several methods for estimating unimodal densities are proposed: plug-in MLE, pregrouping techniques, linear spline MLE. Based on the maximum likelihood method, an automatic procedure for estimating a unimodal density as well as its mode is proposed. We also give asymptotic theory for the proposed estimators. An important consequence of this study is that having to estimate the location of a mode does not affect the limiting behavior of the proposed unimodal density estimate. Simulation studies illustrate the proposed methods.

<div>Without Abstract

Let G=(V,E) be a locally finite connected weighted graph, and Ω be an unbounded subset of V. Using Rothe's method, we study the existence of solutions for the semilinear heat equation ∂_t u+|u|^{p−1}⋅u=Δu (p≥1) and the parabolic variational inequality ∫_{Ω^∘} ∂_t u⋅(v−u)dμ ≥ ∫_{Ω^∘} (Δu+f)⋅(v−u)dμ for any v∈H, where H={u∈W^{1,2}(V):u=0 on V∖Ω^∘}.

We construct gradient K\"ahler-Ricci solitons on Ricci-flat K\"ahler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the results generalize constructions of Cao and Feldman-Ilmanen-Knopf.