A uniform coerciveness result for biharmonic operator and its application to a parabolic equation

We establish an $L^2$ a priori estimate for solutions to the problem: $\Delta^2u=f$ in $\Omega$, $\frac{\partial u}{\partial n}=0$ on $\partial\Omega$, $-\frac{\partial}{\partial n}(\Delta u)+\beta\alpha u=0$ on $\partial\Omega$, where $n$ is the outward unit normal vector to $\partial\Omega$, $\alpha$ is a positive function on $\partial\Omega$ and $\beta$ is a nonnegative parameter. Our estimate is stable under the singular limit $\beta\to\infty$ and cannot be absorbed into the results of S. Agmon, A. Douglis and L. Nirenberg. We apply the estimate to the analysis of the large-time limit of a solution to the equation $(\frac{\partial}{\partial t}+\Delta^2)u(x,t)=f(x,t)$ in an asymptotically cylindrical domain $D$, where we impose a boundary condition similar to that above and the coefficient of $u$ in the boundary condition is supposed to tend to $+\infty$ as $t\to\infty$.