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.
Moments and convex optimiza- tion for analysis and control of nonlinear pdes
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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.