A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies

Alberti, Giovanni; Bellettini, Giovanni

doi:10.1017/s0956792598003453

Cited by 128 publications

(180 citation statements)

References 24 publications

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“…A related result on Gamma-convergence and optimal profiles was obtained by AlbertiBellettini [4] for anisotropic singularities that are integrable, unlike our…”

Section: Perturbation Of the Norm H Ssupporting

confidence: 51%

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Gamma convergence of an energy functional related to the fractional Laplacian

González

2009

Calc. Var.

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We prove a Γ-convergence result for an energy functional related to some fractional powers of the Laplacian operator, (−∆) s for 1/2 < s < 1, with two singular perturbations, that leads to a two-phase problem. The case (−∆) 1/2 was considered by Alberti-Bouchitté-Seppecher in relation to a model in capillarity with line tension effect. However, the proof in our setting requires some new ingredients such as the CaffarelliSilvestre extension for the fractional Laplacian and new trace inequalities for weighted Sobolev spaces.

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“…A related result on Gamma-convergence and optimal profiles was obtained by AlbertiBellettini [4] for anisotropic singularities that are integrable, unlike our…”

Section: Perturbation Of the Norm H Ssupporting

confidence: 51%

“…The proof is essentially the one of Modica and can be found in Alberti [3]. By a well known truncation argument ( [4], lemma 1.14), we can assume that u : A → [α, β]. Now, use the inequality…”

Section: γ-Convergence In the Interiormentioning

confidence: 99%

Gamma convergence of an energy functional related to the fractional Laplacian

González

2009

Calc. Var.

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“…is finite only if u is a function of bounded variation and u ∈ {a, b} a.e., and in that case it is given by (1.1), with A variant proposed for the Cahn-Hilliard functional F ε reads as follows (see [15], [1], [2]): This corresponds to the replacement of the Dirichlet term ε 2 Ω |∇u| 2 in G ε with a non-local term where spatial inhomogeneity, weighted with an influence kernel, is penalized.…”

Section: Introductionmentioning

confidence: 99%

Variational models for phase separation

Solci

Vitali

2003

Interfaces Free Bound.

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The paper deals with the asymptotic behaviour (as ε → 0) of a family F ε (u, v) of integral functionals in the framework of phase separation. In order to obtain a selection criterion for the minima of the usual double-well, non-convex free energy involving the phase-variable u, we add a gradient term in a new variable v which is related to u through the L 2 -distance between u and v, weighted by a coefficient α. We prove that the limit as ε → 0 is a minimal area model with a surface tension of non-local form. The well-known Modica-Mortola constant can be recovered in this setting as a limit case when α → +∞.

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“…In this way the non-uniqueness problem for E 0 is resolved. Rigorous results on this topic can be found in [4]. …”

Section: The Energy Functional For the Equilibriummentioning

confidence: 99%

Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

Rohde

2005

Interfaces Free Bound.

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Abstract. The one-dimensional system of elasticity with a non-monotone or convex-concave stress-strain relation provides a model to describe the longitudinal dynamics of solid-solid phase transitions in a bar. If dissipative effects are neglected it takes the form of a system of first-order nonlinear conservation laws and dynamical phase boundaries appear as shock wave solutions. In the physically most relevant cases these shocks are of the non-classical undercompressive type and therefore entropy solutions of the associated Cauchy problem are not uniquely determined. Important dissipative effects that lead to unique regular solutions are viscosity and capillarity where the latter effect is usually modelled by at least third-order spatial derivatives. Differently from these models we consider a novel type of non-local regularization that models both effects but avoids high-order derivatives. We suggest a particular scaling for the dissipative terms and conjecture that with this scaling the regular solutions single out unique physically relevant weak solutions of the first-order conservation law in the limit of vanishing dissipation parameter. We verify the conjecture first by proving that the non-local system admits special solutions of traveling-wave type that correspond to dynamical phase boundaries. Moreover it is proven that regular solutions of a general Cauchy problem converge to weak solutions of the system of first-order conservation laws. The proof is achieved by the method of compensated compactness.

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A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies

Cited by 128 publications

References 24 publications

Gamma convergence of an energy functional related to the fractional Laplacian

Gamma convergence of an energy functional related to the fractional Laplacian

Variational models for phase separation

Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

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