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2009, 2009(Special): 333-339. doi: 10.3934/proc.2009.2009.333

$H^2$-solutions for some elliptic equations with nonlinear boundary conditions

Junichi Harada 1, and  Mitsuharu Ôtani 2,

1. 

Major in Pure and Applied Physics, Graduate School of Advanced Sciences and Engineering, Waseda University, Japan
2. 

Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555

Received  August 2008 Revised  June 2009 Published  September 2009

The following elliptic equation with nonlinear boundary condition is considered: $-\Delta u+bu=f(x)$ in $\Omega$, $-\frac{\partial u}{\partial n}=\beta(u)-g(u)$ on $\partial\Omega$, where $b\geq0$, $f\in L^2(\Omega)$, $\beta(u)$ is a monotone increasing function on $\mathbb (R)^1$ and $g(u)$ is its small perturbation. It is shown that this problem admits a solution $u$ belonging to $H^2(\Omega)$ under suitable conditions on $\beta$ and $g$. The method of our proof relies on some approximation procedures and the classical but new arguments for $H^2$-estimates near the boundary which can work under (non-monotone) nonlinear boundary conditions.
Keywords: Nonlinear boundary condition, $H^2$-estimates.
Mathematics Subject Classification: Primary: 35A.
Citation: Junichi Harada, Mitsuharu Ôtani. $H^2$-solutions for some elliptic equations with nonlinear boundary conditions. Conference Publications, 2009, 2009 (Special) : 333-339. doi: 10.3934/proc.2009.2009.333
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