Abstract
AbstractWe consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-Dirichlet energy in this class. We find a quantity $$\lambda [f]$$
λ
[
f
]
which controls the symmetry, uniqueness and regularity of minimisers: if $$\lambda [f]\le 1$$
λ
[
f
]
≤
1
then minimisers are symmetric and unique; if $$\lambda [f]$$
λ
[
f
]
is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if $$\lambda [f]$$
λ
[
f
]
is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11(3):275–296, 1994). Some of our results also extend to the setting where the ball is replaced by $${\mathbb {R}}^2$$
R
2
and boundary conditions are not prescribed.
Funder
Engineering and Physical Sciences Research Council
European Research Council
Publisher
Springer Science and Business Media LLC
Subject
Mechanical Engineering,Mathematics (miscellaneous),Analysis
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