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 Mosc. Math. J., 2016, Volume 16, Number 1, Pages 179–200 (Mi mmj597)

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

Kazushi Yoshitomi

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minamiohsawa 1-1, Hachioji, Tokyo 192-0397, Japan

Abstract: 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 $t\to\infty$.

Key words and phrases: biharmonic operator, singular perturbation, parabolic equation, stabilization.

DOI: https://doi.org/10.17323/1609-4514-2016-16-1-179-200

Full text: http://www.mathjournals.org/.../2016-016-001-006.html
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Bibliographic databases:

MSC: 35J35, 35J40, 35K35
Received: October 31, 2013; in revised form January 23, 2015
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Citation: Kazushi Yoshitomi, “A uniform coerciveness result for biharmonic operator and its application to a parabolic equation”, Mosc. Math. J., 16:1 (2016), 179–200

Citation in format AMSBIB
\Bibitem{Yos16} \by Kazushi~Yoshitomi \paper A uniform coerciveness result for biharmonic operator and its application to a~parabolic equation \jour Mosc. Math.~J. \yr 2016 \vol 16 \issue 1 \pages 179--200 \mathnet{http://mi.mathnet.ru/mmj597} \crossref{https://doi.org/10.17323/1609-4514-2016-16-1-179-200} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3470579} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000336402400008}