Phragmén-Lindelöf type results for harmonic functions with nonlinear boundary conditions
The subject of this study are harmonic functions \(u\) in a semi-infinite cylinder \(\{(x_ 1,x_ 2,x_ 3):(x_ 1,x_ 2) \in D\}\) with prescribed Dirichlet and Neumann conditions on \(D\) and nonlinear boundary conditions \(\partial u/ \partial n+f(u)=0\) on \(\partial D \times \mathbb{R}^ +\). No conditions are given at infinity. It is shown that the \(L_ 2\)- norm \[ \varphi(z)=\int_ Du^ 2 (x_ 1,x_ 2,z) dx_ 1 dx_ 2 \] decays or grows exponentially or polynomially according to the growth of the nonlinearity \(f\). The proof is based on a differential inequality for \(\varphi (z)\). This inequality is obtained by means of elementary arguments, however it requires a great skill and knowledge of the structure of harmonic functions.
scientific article; zbMATH DE number 410078
scientific article