Bulletin of the Belgian Mathematical Society - Simon Stevin

Elliptic patching of harmonic functions

Cristina Giannotti

Full-text: Open access

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

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

Keywords
Patching of harmonic functions maximum principle for solutions to elliptic equations

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


Export citation