Lina Wu and Jia Liu and Ye Li
Discovering LiouvilleType Problems for pEnergy Minimizing Maps in Closed HalfEllipsoids by Calculus Variation Method
527 - 533
2016
10
10
International Journal of Mathematical and Computational Sciences
https://publications.waset.org/pdf/10006222
https://publications.waset.org/vol/118
World Academy of Science, Engineering and Technology
The goal of this project is to investigate constant
properties (called the Liouvilletype Problem) for a pstable map
as a local or global minimum of a penergy functional where
the domain is a Euclidean space and the target space is a
closed halfellipsoid. The First and Second Variation Formulas
for a penergy functional has been applied in the Calculus
Variation Method as computation techniques. Stokes&rsquo; Theorem,
CauchySchwarz Inequality, HardySobolev type Inequalities, and
the Bochner Formula as estimation techniques have been used to
estimate the lower bound and the upper bound of the derived
pHarmonic Stability Inequality. One challenging point in this project
is to construct a family of variation maps such that the images
of variation maps must be guaranteed in a closed halfellipsoid.
The other challenging point is to find a contradiction between the
lower bound and the upper bound in an analysis of pHarmonic
Stability Inequality when a penergy minimizing map is not constant.
Therefore, the possibility of a nonconstant penergy minimizing
map has been ruled out and the constant property for a penergy
minimizing map has been obtained. Our research finding is to explore
the constant property for a pstable map from a Euclidean space into
a closed halfellipsoid in a certain range of p. The certain range of
p is determined by the dimension values of a Euclidean space (the
domain) and an ellipsoid (the target space). The certain range of p
is also bounded by the curvature values on an ellipsoid (that is, the
ratio of the longest axis to the shortest axis). Regarding Liouvilletype
results for a pstable map, our research finding on an ellipsoid is a
generalization of mathematicians&rsquo; results on a sphere. Our result is
also an extension of mathematicians&rsquo; Liouvilletype results from a
special ellipsoid with only one parameter to any ellipsoid with (n1)
parameters in the general setting.
Open Science Index 118, 2016