A priori estimates for finite-energy sequences of one-dimensional Cahn-Hilliard functional with non-standard multi-well potential

Andrija Raguž

Vol. 29 No. 1 (2024)

Regular Articles

A priori estimates for finite-energy sequences of one-dimensional Cahn-Hilliard functional with non-standard multi-well potential

Published 2024-03-12

Andrija Raguž

Andrija Raguž
Zagreb School of Economics and Management

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Abstract

In this paper we provide some results pertaining to asymptotic behaviour as $\varepsilon\str 0$ of the finite-energy sequences of the one-dimensional Cahn-Hilliard functional $$I^{\varepsilon}_0(u)=\int_{0}^{1}\Big({\varepsilon}^2u'^2(s)+W(u(s))\Big)ds,$$ where $u\in {\rm H}^{1}\oi{0}{1}$ and where $W$ is a multi-well potential endowed with a non-standard integrability condition. We introduce a new class of finite-energy sequences, we recover its underlying geometric properties as $\vrepsilon\str 0$, and obtain the related a priori estimates.

Keywords

asymptotic analysis, singular perturbation, Young measures, Cahn-Hiliard functional, regularity

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Author Biography

Andrija Raguž

Department of Economics and Mathematics, full professor.