These notes have been motivated by the interests of the author in variational problems depending on small parameters, for some of which a description based on a global minimization principle does not seem satisfactory. Such problems range from the derivation of physical theories from first principles to numerical problems involving energies with many local minima. Even though an asymptotic description of related global minimization problems can be given in terms of Gamma-convergence, the Gamma-limit often does not capture the behavior of local minimizers or of gradient flows. This failure is sometimes mentioned as the proof that Gamma-convergence is `wrong'. It may well be so. The author's standpoint is that it might nevertheless be a good starting point that may be systematically `corrected'. The author's program has been to examine the (few) results in the literature, and try to connect them with his own work in homogenization and discrete systems, where often the local minimization issues are crucial due to the oscillations of the energies. The directions of research have been 1) find criteria that ensure the convergence of local minimizers and critical points. In case this does not occur then modify the Gamma-limit into an equivalent Gamma-expansion (as introduced by the author and L. Truskinovsky) in order to match this requirement. We note that in this way we `correct' some limit theories, finding (or `validating') other ones present in the literature; 2) modify the concept of local minimizer, so that it may be more `compatible' with the process of Gamma-limit. One such concept is the epsilon-stability of C. Larsen; 3) treat evolution problems for energies with many local minima obtained by a timediscrete scheme (introducing the notion of `minimizing movements along a sequence of functionals'). In this case the minimizing movement of the Gamma-limit can be always obtained by a choice of the space and time-scale, but more interesting behaviors can be obtained at a critical ratio between them. In many cases a `critical scale' can be computed and an effective motion, from which all other minimizing movements are obtained by scaling. Furthermore the choice of suitable Gamma-converging sequences in the scheme above allows to address the issues of long-time behavior and backwards motion; 4) examine the general variational evolution results that may be related to these minimizing movements, in particular recent theories of quasistatic motion and gradent flow in metric spaces.

Braides, A. (2014). Local minimization, variational evolution and Gamma-convergence. Springer.

Local minimization, variational evolution and Gamma-convergence

BRAIDES, ANDREA
2014-01-01

Abstract

These notes have been motivated by the interests of the author in variational problems depending on small parameters, for some of which a description based on a global minimization principle does not seem satisfactory. Such problems range from the derivation of physical theories from first principles to numerical problems involving energies with many local minima. Even though an asymptotic description of related global minimization problems can be given in terms of Gamma-convergence, the Gamma-limit often does not capture the behavior of local minimizers or of gradient flows. This failure is sometimes mentioned as the proof that Gamma-convergence is `wrong'. It may well be so. The author's standpoint is that it might nevertheless be a good starting point that may be systematically `corrected'. The author's program has been to examine the (few) results in the literature, and try to connect them with his own work in homogenization and discrete systems, where often the local minimization issues are crucial due to the oscillations of the energies. The directions of research have been 1) find criteria that ensure the convergence of local minimizers and critical points. In case this does not occur then modify the Gamma-limit into an equivalent Gamma-expansion (as introduced by the author and L. Truskinovsky) in order to match this requirement. We note that in this way we `correct' some limit theories, finding (or `validating') other ones present in the literature; 2) modify the concept of local minimizer, so that it may be more `compatible' with the process of Gamma-limit. One such concept is the epsilon-stability of C. Larsen; 3) treat evolution problems for energies with many local minima obtained by a timediscrete scheme (introducing the notion of `minimizing movements along a sequence of functionals'). In this case the minimizing movement of the Gamma-limit can be always obtained by a choice of the space and time-scale, but more interesting behaviors can be obtained at a critical ratio between them. In many cases a `critical scale' can be computed and an effective motion, from which all other minimizing movements are obtained by scaling. Furthermore the choice of suitable Gamma-converging sequences in the scheme above allows to address the issues of long-time behavior and backwards motion; 4) examine the general variational evolution results that may be related to these minimizing movements, in particular recent theories of quasistatic motion and gradent flow in metric spaces.
Settore MAT/05 - Analisi Matematica
English
Rilevanza internazionale
Monografia
Braides, A. (2014). Local minimization, variational evolution and Gamma-convergence. Springer.
Monografia
Braides, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/121450
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