Boundary decay estimates for solutions of fourth-order elliptic equations

On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic syste…

Spectral asymptotics for a singularly perturbed fourth order locally periodic self-adjoint elliptic operator

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective opera…

四阶边值问题的谱数据渐近性质

Spectral data asymptotics for fourth-order boundary value problems

本文推导了具有分布系数和三种分离边界条件的四阶线性微分方程谱数据的精确渐近性质，方法依赖高阶微分算子正则化与渐近分析，结果应用于逆谱问题理论并具独立意义。

On the sign-regularity of positive fourth-order differential operators

It is shown that positivity in $(0,1)\times (0,1)$ of Green function of positively defined fourth-order ordinary differential operator (with separated boundary conditions) is a criterium of sign-regularity of this operator.