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.
Solving the plateau’s problem for hypersurfaces without the compact- ness theorem for integral currents.Geometric measure theory and the calculus of variations, 44:255–295
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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.