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

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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

Abstract
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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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