Abstract
Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L3 as the nontrivial zero mean curvature surfaces which also satisfy the harmonic equation, the wave equation, and a degenerate equation which is derived from the harmonic equation or the wave equation. This elementary and analytic result shows that the change of the roles of dependent and independent variables may be useful in solving differential equations.
| Original language | English |
|---|---|
| Pages (from-to) | 666-670 |
| Number of pages | 5 |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 353 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 2009 May 15 |
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Keywords
- Harmonic graph
- Minimal surfaces
- Wave graph
- Zero mean curvature surfaces
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
Cite this
Minimal harmonic graphs and their Lorentzian cousins. / Kim, Young Wook; Lee, Hyung Yong; Yang, Seong-Deog.
In: Journal of Mathematical Analysis and Applications, Vol. 353, No. 2, 15.05.2009, p. 666-670.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Minimal harmonic graphs and their Lorentzian cousins
AU - Kim, Young Wook
AU - Lee, Hyung Yong
AU - Yang, Seong-Deog
PY - 2009/5/15
Y1 - 2009/5/15
N2 - Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L3 as the nontrivial zero mean curvature surfaces which also satisfy the harmonic equation, the wave equation, and a degenerate equation which is derived from the harmonic equation or the wave equation. This elementary and analytic result shows that the change of the roles of dependent and independent variables may be useful in solving differential equations.
AB - Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L3 as the nontrivial zero mean curvature surfaces which also satisfy the harmonic equation, the wave equation, and a degenerate equation which is derived from the harmonic equation or the wave equation. This elementary and analytic result shows that the change of the roles of dependent and independent variables may be useful in solving differential equations.
KW - Harmonic graph
KW - Minimal surfaces
KW - Wave graph
KW - Zero mean curvature surfaces
UR - http://www.scopus.com/inward/record.url?scp=58549103690&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=58549103690&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2008.12.025
DO - 10.1016/j.jmaa.2008.12.025
M3 - Article
VL - 353
SP - 666
EP - 670
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 2
ER -