Limit Shapes of Local Minimizers for the Alt–Caffarelli Energy Functional in Inhomogeneous Media

2020 ◽

pp. 1-29 ◽

Cited By ~ 2

Author(s):

Dario Pierotti ◽

Nicola Soave ◽

Gianmaria Verzini

Keyword(s):

State Energy ◽

Ground States ◽

Energy Functional ◽

Negative Energy ◽

Topological Properties ◽

Local Minimizers ◽

Metric Graphs ◽

Global Minimizers ◽

Underlying Graph ◽

Mass Constraint

We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

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Count and Symmetry of Global and Local Minimizers of the Cahn-Hilliard Energy Over Cylindrical Domains

Advanced Nonlinear Studies ◽

10.1515/ans-2012-0108 ◽

2012 ◽

Vol 12 (1) ◽

Cited By ~ 1

Author(s):

Arnaldo Simal do Nascimento ◽

João Biesdorf ◽

Janete Crema

Keyword(s):

Three Dimensional ◽

Unique Continuation ◽

Energy Functional ◽

Circular Cylinders ◽

Cylindrical Domain ◽

Local Minimizers ◽

Global Minimizers ◽

Cylindrical Domains ◽

Γ Convergence ◽

Global And Local

AbstractWe address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion.The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

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AN ADAPTIVE FINITE ELEMENT APPROXIMATION OF A GENERALIZED AMBROSIO–TORTORELLI FUNCTIONAL

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s021820251350019x ◽

2013 ◽

Vol 23 (09) ◽

pp. 1663-1697 ◽

Cited By ~ 33

Author(s):

SIOBHAN BURKE ◽

CHRISTOPH ORTNER ◽

ENDRE SÜLI

Keyword(s):

Finite Element ◽

Finite Element Approximation ◽

Energy Functional ◽

Adaptive Finite Element Method ◽

Element Approximation ◽

Adaptive Finite Element ◽

Local Minimizers ◽

Constrained Minimization Problem ◽

Adaptive Finite Element Approximation ◽

Bound Constrained Minimization

The Francfort–Marigo model of brittle fracture is posed in terms of the minimization of a highly irregular energy functional. A successful method for discretizing the model is to work with an approximation of the energy. In this work a generalized Ambrosio–Tortorelli functional is used. This leads to a bound-constrained minimization problem, which can be posed in terms of a variational inequality. We propose, analyze and implement an adaptive finite element method for computing (local) minimizers of the generalized functional.

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On the convexity condition for the Semi-Geostrophic system

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2021018 ◽

2021 ◽

Author(s):

Adrian Tudorascu

Keyword(s):

Vector Field ◽

Critical Points ◽

Vector Fields ◽

Energy Functional ◽

Convexity Condition ◽

Local Minimizers ◽

Flow Maps ◽

Divergence Free Vector ◽

Absolute Density ◽

The Absolute

We show that conservative distributional solutions to the Semi-Geostrophic systems in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge- Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admis- sible vector field. The obligatory connection with the Monge-Kantorovich theory is addressed briefly.

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