The classical and relaxed minima coincide for codomain dimension one in calculus of variations and optimal control, but a positive gap can exist when both domain and codomain dimensions exceed one.
Unrectifiable normal currents in Euclidean spaces
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abstract
We construct in $\mathbb{R}^{k+2}$ a $k$-dimensional simple normal current whose support is purely $2$-unrectifiable. The result is sharp because the support of a normal current cannot be purely $1$-unrectifiable and a $(k+1)$-dimensional normal current can be represented as an integral of $(k+1)$-rectifiable currents. This gives a negative answer to the (revised version) of a question of Frank Morgan (1984).
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2022 1verdicts
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The gap between a variational problem and its occupation measure relaxation
The classical and relaxed minima coincide for codomain dimension one in calculus of variations and optimal control, but a positive gap can exist when both domain and codomain dimensions exceed one.