From gradient theory of phase transition to a generalized minimal interface problem with a contact energy

Tien-Tsan  Shieh

doi:10.3934/dcds.2016.36.2729

Article Contents

2016, Volume 36, Issue 5: 2729-2755. Doi: 10.3934/dcds.2016.36.2729

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From gradient theory of phase transition to a generalized minimal interface problem with a contact energy

Tien-Tsan Shieh^1,

1.
Department of Mathematics, National Central University, Chung-Li 320

Received: May 31, 2015

Revised: June 30, 2015

Published: May 2017

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Abstract

We consider asymptotic behaviours of a variational problem $$ \inf_{u\in \mathcal A(m,f)} \int_\Omega \left\{\frac{\epsilon^2}{2} \left|\nabla u\right|^2 + \frac{V(x)}{2}u^2 + \frac{1}{4}u^4\right\}\,dx$$ over a admissible class $\mathcal A(m,f)=\{u\in W^{1,2}(\Omega):\,\int_\Omega u^2\,dx=m,\,u=f \textrm{ on }\partial \Omega\}$. The problem demonstrates some features of the phase separation in experimental studies of Bose-Einstein condensation confined in an infinite-trap potential. In this paper, we show the limiting variational problem is a generalized minimal interface problem involving a boundary contact energy. The asymptotic behaviour of the minimizers $\{u_\epsilon\}$ is characterized by a generalized mean curvature equation and a contact angle relation, the Young's relation, at the junction of the interfaces and the boundary. An example is given to demonstrate the possible existence of local minimizers $\{v_\epsilon\}_{\epsilon>0}$ for the perturbed variational problem due to suitable Dirichlet boundary condition $u=f$.

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Mathematics Subject Classification: Primary: 49J45, 35J25, 35J75; Secondary: 35Q40, 49S05.

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