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Glasnik matematički, Vol. 59 No. 1, 2024.

Original scientific paper

https://doi.org/10.3336/gm.59.1.06

Some results in asymptotic analysis of finite-energy sequences of one-dimensional Cahn–Hilliard functional with non-standard two-well potential

Andrija Raguž orcid id orcid.org/0000-0001-8045-9636 ; Department of Economics and Mathematics, Zagreb School of Economics and Management, Filipa Vukasovića 1, 10 000 Zagreb, Croatia

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page 125-145

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Abstract

In this paper we extend the consideration of G. Leoni
pertaining to the finite-energy sequences of the one-dimensional
Cahn-Hilliard functional
\[
I^{\varepsilon}_0(u)=\int_{0}^{1}\Big({\varepsilon}^2
u'^2(s)+W(u(s))\Big)ds,
\]
where \(u\in {\rm H}^{1}(0,1)\) and where \(W\) is a two-well
potential with symmetrically placed wells endowed with a
non-standard integrability condition. We introduce several new
classes of finite-energy sequences, we recover their underlying geometric properties as
\(\varepsilon\longrightarrow 0\), and we prove the related compactness result.

Keywords

Asymptotic analysis, singular perturbation, Young measures, Cahn-Hilliard functional, compactness

Hrčak ID:

318144

URI

https://hrcak.srce.hr/318144

Publication date:

25.11.2024.

Visits: 273 *

Publication date: 20 June 2024

Volume: Vol 59

Issue: Svezak 1

Pages: 125-145

DOI: 10.3336/gm.59.1.06

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: Asymptotic analysis, singular perturbation, Young measures, Cahn-Hilliard functional, compactness

Some results in asymptotic analysis of finite-energy sequences of one-dimensional Cahn–Hilliard functional with non-standard two-well potential

Andrija Raguž[1]

Email: araguz@zsem.hr

 

[1] Department of Economics and Mathematics, Zagreb School of Economics and Management, Filipa Vukasovića 1, 10 000 Zagreb, Croatia

Abstract

In this paper we extend the consideration of G. Leoni pertaining to the finite-energy sequences of the one-dimensional Cahn-Hilliard functional \[ I^{\varepsilon}_0(u)=\int_{0}^{1}\Big({\varepsilon}^2 u'^2(s)+W(u(s))\Big)ds, \] where \(u\in {\rm H}^{1}(0,1)\) and where \(W\) is a two-well potential with symmetrically placed wells endowed with a non-standard integrability condition. We introduce several new classes of finite-energy sequences, we recover their underlying geometric properties as \(\varepsilon\longrightarrow 0\), and we prove the related compactness result.

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