$H^2$-solutions for some elliptic equations with nonlinear boundary conditions
Junichi Harada - Major in Pure and Applied Physics, Graduate School of Advanced Sciences and Engineering, Waseda University, 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
Received: August 2008; Revised: June 2009; Published: September 2009. |