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

Junichi Harada; Mitsuharu Ôtani

doi:10.3934/proc.2009.2009.333

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2009, Volume 2009: 333-339. Doi: 10.3934/proc.2009.2009.333 open access

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$H^2$-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
2.
Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555

Received: July 31, 2008

Revised: May 31, 2009

Published: August 2009

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Abstract

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.

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