## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Elliptic patching of harmonic functions

Cristina Giannotti

#### 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 2 (2008), 257-268.

Dates
First available in Project Euclid: 8 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1210254823

Digital Object Identifier
doi:10.36045/bbms/1210254823

Mathematical Reviews number (MathSciNet)
MR2424111

Zentralblatt MATH identifier
1160.31001

#### Citation

Giannotti, Cristina. Elliptic patching of harmonic functions. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 2, 257--268. doi:10.36045/bbms/1210254823. https://projecteuclid.org/euclid.bbms/1210254823