Monday Analysis Seminar: A uniform coerciveness result for biharmonic operator and its application to a parabolic equation

2013-10-28 14:45 - 2013-10-28 16:30
Faculty of Science Buliding #3 Room 210
Kazushi Yoshitomi (Tokyo Metropolitan University)
We establish an $L^{2}$ a priori estimate for solutions to the problem
\frac{\partial u}{\partial n}=0\quad\mbox{on}\quad\partial\Omega,\\
-\frac{\partial}{\partial n}(\Delta u)+\beta\alpha u=0\quad\mbox{on}\quad\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 (Comm. Pure Appl. Math. 12 (1959), 623--727).  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 $t\to\infty$.  Our work here is primarily motivated by that of A. Friedman (Acta Math. 106 (1961), 1--43).