Regularity and relaxed problems of minimizing biharmonic maps into spheres
Abstract
For
n≥5
and
k≥4
, we show that any minimizing biharmonic map from
Ω⊂
R
n
to
S
k
is smooth off a closed set whose Hausdorff dimension is at most
n−5
. When
n=5
and
k=4
, for a parameter
λ∈[0,1]
we introduce a
λ
-relaxed energy $\H_\lambda$ for the Hessian energy for maps in
W
2,2
(Ω,
S
4
)
so that each minimizer
u
λ
of $\H_\lambda$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_\lambda$ for
λ∈[0,1)
.