Asymptotics of the solution to Robin problem
The authors consider the following problem: \[ Au:=\nabla^2u=f \text{ on }\mathbb{D}',\;u(\infty)=0\quad\frac{\partial u}{\partial\vec N}-\zeta u=0\text{ on }\Gamma,\tag{1} \] where \(\mathbb{D}\subset\mathbb{R}^3\) is a bounded domain with \(C^\infty\)-boundary \(\Gamma\); \(\mathbb{D}'=\mathbb{R}^3 \setminus\mathbb{D}\), \(\vec N\) is an outer unit normal to \(\Gamma\), \(f\in L^2_0(\mathbb{D}')\), \(\zeta\) is a constant, \(\text{Im}\,\zeta\neq 0\), or \(\text{Im}\,\zeta=0\) and then \(\zeta\geq 0\). The authors adress to the following questions: a) Does \(u\) converge in some sense to the solution of the problem (2) as \(|\zeta|\to\infty\)? \[ \nabla^2w=f \text{ in }\mathbb{D}',\;w(\infty)=0\quad w=0\text{ on }\Gamma\tag{2} \] b) At what rate does \(u\) converge to \(w\) as \(|\zeta|\to\infty\)?
scientific article; zbMATH DE number 5362142
scientific article