The effect of a singular perturbation on nonconvex variational problems

Sternberg, Peter

doi:10.1007/bf00253122

Cited by 362 publications

(346 citation statements)

References 9 publications

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“…The physicists [11] actually considered potentials U in the soliton bag model of the form U : φ → W (φ) + b|φ| 2 satisfying the conditions of Theorem 1.16. [15] studying the dependence of the numerical solutions on the parameters exhibit behaviors of the φ field similar to the ones of the Modica-Mortolla problem [29,30,28,36,3]. Nevertheless, this is the first result which shows clearly the link between the two models we studied.…”

Section: Theorem 116 Assume That the Condition Of Proposition 115 mentioning

confidence: 61%

“…This proposition is an adaptation of the result of Modica and Mortola [29,30] generalized by Modica [28] (see also Sternberg [36] or Braides [3]) for the gradient theory of phase transitions in an unbounded setting. Its proof strongly uses the one of [36].…”

Section: The Bag Approximation Model As a γ-Limit Of Soliton Bag Modelsmentioning

confidence: 65%

“…These are the first rigorous proofs of existence for these two models. Actually, solving the bag approximation model is a shape optimization problem on finite perimeter sets of R 3 which is related to the soliton bag model thanks to the gradient theory of phase transitions [29,30,28,36,3,4]. Indeed, we show that the energy functionals of the bag approximation model are Γ-limits of sequences of soliton bag model energy functionals for the ground and excited state problems.…”

Section: Introduction and Main Resultsmentioning

confidence: 88%

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A variational study of some hadron bag models

Treust

2013

Calc. Var.

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Abstract. We study, in this paper, some relativistic hadron bag models. We prove the existence of excited state solutions in the symmetric case and of a ground state solution in the non-symmetric case for the soliton bag and the bag approximation models by concentration compactness. We show that the energy functionals of the bag approximation model are Γ-limits of sequences of soliton bag energy functionals for the ground and excited state problems. The pre-compactness, up to translation, of the sequence of ground state solutions associated with the soliton bag energy functionals in the non-symmetric case is obtained combining the Γ-convergence theory and the concentrationcompactness principle. Finally, we give a rigorous proof of the original derivation of the M.I.T. bag equations via a limit of bag approximation ground state solutions in the spherical case. The supersymmetry property of the Dirac operator is a key point in many of our arguments.

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Section: Theorem 116 Assume That the Condition Of Proposition 115 mentioning

confidence: 61%

Section: The Bag Approximation Model As a γ-Limit Of Soliton Bag Modelsmentioning

confidence: 65%

Section: Introduction and Main Resultsmentioning

confidence: 88%

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A variational study of some hadron bag models

Treust

2013

Calc. Var.

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“…Modica [21] and Sternberg [40] proved that minimizers of E ε under a volume constraint converge to areaminimizing hypersurfaces with an integral constraint. Luckhaus-Modica [19] then showed that the Lagrange-multipliers associated with the volume constraint converge to the constant mean curvature of the limiting hypersurface.…”

Section: Related Results and Main Techniquesmentioning

confidence: 99%

“…The behavior of (locally) energy minimizing solutions of (7.2) is well understood [16,19,21,40]. In this case sequences (u ε ) ε>0 with uniformly bounded energy converge to a constant-mean curvature hypersurface with single-multiplicity.…”

Section: Proposition 611mentioning

confidence: 99%

Convergence of phase-field approximations to the Gibbs–Thomson law

Röger

Tonegawa

2007

Calc. Var.

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We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van der Waals-CahnHilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta-Kawasaki as a model for micro-phase separation in block-copolymers.

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On the convergence of stable phase transitions

Padilla

Tonegawa

1998

Comm. Pure Appl. Math.

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The effect of a singular perturbation on nonconvex variational problems

Cited by 362 publications

References 9 publications

A variational study of some hadron bag models

A variational study of some hadron bag models

Convergence of phase-field approximations to the Gibbs–Thomson law

On the convergence of stable phase transitions

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