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

20131028 14:45 
20131028 16:30
 Place
 Faculty of Science Buliding #3 Room 210

Speaker/Organizer
 Kazushi Yoshitomi (Tokyo Metropolitan University)

 We establish an $L^{2}$ a priori estimate for solutions to the problem
\[
\left\{
\begin{array}{l}
\Delta^{2}u=f\quad\mbox{in}\quad\Omega,\\
\frac{\partial u}{\partial n}=0\quad\mbox{on}\quad\partial\Omega,\\
\frac{\partial}{\partial n}(\Delta u)+\beta\alpha u=0\quad\mbox{on}\quad\partial\Omega,
\end{array}
\right.
\]
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), 623727). We apply the estimate to the analysis of the largetime 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), 143).