Abstract
Given two harmonic functions $u_{+}(x,y)$, $u_{-}(x,y)$ defined on opposite sides of the $y$-axis in $\mathbb{R}^2$ and periodic in $y$, we consider the problem of constructing a {\it family of gluing elliptic functions}, i.e. a family of functions $u_{\epsilon}(x,y)$ of class ${\mathcal C}^{1,1}$ that coincide with $u_+$ and $u_-$ outside neighborhoods of the $y$-axis of width less than $\epsilon$ and are solutions to linear, uniformly elliptic equations without zero order terms. We first show that not always there is such a family and we give a necessary condition for its existence. Then we give a sufficient condition for the existence of a family of gluing elliptic functions and a way for its construction.
Citation
Cristina Giannotti. "Elliptic patching of harmonic functions." Bull. Belg. Math. Soc. Simon Stevin 15 (2) 257 - 268, May 2008. https://doi.org/10.36045/bbms/1210254823
Information