Abstract
We study the Γ-convergence of the following functional (p > 2)
where Ω is an open bounded set of \({\mathbb{R}^3}\) and W and V are two non-negative continuous functions vanishing at α, β and α′, β′, respectively. In the previous functional, we fix a = 2 − p and u is a scalar density function, Tu denotes its trace on ∂Ω, d(x, ∂Ω) stands for the distance function to the boundary ∂Ω. We show that the singular limit of the energies \({F_{\varepsilon}}\) leads to a coupled problem of bulk and surface phase transitions.
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Palatucci, G., Sire, Y. Γ-Convergence of some super quadratic functionals with singular weights. Math. Z. 266, 533–560 (2010). https://doi.org/10.1007/s00209-009-0584-x
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DOI: https://doi.org/10.1007/s00209-009-0584-x
Keywords
- Phase transitions
- Line tension
- Weighted Sobolev spaces
- Nonlocal variational problems
- Γ-Convergence
- Functions of bounded variation