Bulletin of the Belgian Mathematical Society - Simon Stevin

Elliptic patching of harmonic functions

Cristina Giannotti

Full-text: Open access


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

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

First available in Project Euclid: 8 May 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J15: Second-order elliptic equations
Secondary: 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx]

Patching of harmonic functions maximum principle for solutions to elliptic equations


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

Export citation