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Mathematics
Volume 13
Issue 3
10.3390/math13030426
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Article

Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity

by
Marco Degiovanni
*,† and
Marco Marzocchi
†
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via della Garzetta 48, 25133 Brescia, Italy
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2025, 13(3), 426; https://doi.org/10.3390/math13030426
Submission received: 31 October 2024 / Revised: 9 December 2024 / Accepted: 22 January 2025 / Published: 27 January 2025
(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)
Versions Notes

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.
Keywords: calculus of variations; direct methods; invariance by diffeomorphism; quasilinear elliptic equations calculus of variations; direct methods; invariance by diffeomorphism; quasilinear elliptic equations

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MDPI and ACS Style

Degiovanni, M.; Marzocchi, M. Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity. Mathematics 2025, 13, 426. https://doi.org/10.3390/math13030426

AMA Style

Degiovanni M, Marzocchi M. Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity. Mathematics. 2025; 13(3):426. https://doi.org/10.3390/math13030426

Chicago/Turabian Style

Degiovanni, Marco, and Marco Marzocchi. 2025. "Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity" Mathematics 13, no. 3: 426. https://doi.org/10.3390/math13030426

APA Style

Degiovanni, M., & Marzocchi, M. (2025). Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity. Mathematics, 13(3), 426. https://doi.org/10.3390/math13030426

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