We consider the minimization of a functional of the calculus of variations, under assumptions that are diffeomorphism invariant. In particular, a nonuniform coercivity condition needs to be considered. We show that the direct methods of the calculus of variations can be applied in a generalized Sobolev space, which is in turn diffeomorphism invariant. Under a suitable (invariant) assumption, the minima in this larger space belong to a usual Sobolev space and are bounded.
Degiovanni, M., Marzocchi, M., Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity, <<MATHEMATICS>>, 2025; 13 (3): 1-23. [doi:10.3390/math13030426] [https://hdl.handle.net/10807/309416]
Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity
Degiovanni, Marco
Primo
;Marzocchi, MarcoSecondo
2025
Abstract
We consider the minimization of a functional of the calculus of variations, under assumptions that are diffeomorphism invariant. In particular, a nonuniform coercivity condition needs to be considered. We show that the direct methods of the calculus of variations can be applied in a generalized Sobolev space, which is in turn diffeomorphism invariant. Under a suitable (invariant) assumption, the minima in this larger space belong to a usual Sobolev space and are bounded.
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