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

Pages: 333 - 339, Issue Special, September 2009

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Junichi Harada - Major in Pure and Applied Physics, Graduate School of Advanced Sciences and Engineering, Waseda University, Japan (email)
Mitsuharu Ôtani - Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555, Japan (email)

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

Received: August 2008;      Revised: June 2009;      Published: September 2009.