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A singular perturbation result involving the \(H^{1/2}\) norm. (Un résultat de perturbations singulières avec la norme \(H^{1/2}\).) (French. Abridged English version) Zbl 0845.49008

Summary: Let \(I\) be a bounded interval of \(\mathbb{R}\) and \(W\) a continuous non-negative function vanishing only at \(\alpha, \beta\in \mathbb{R}\). We obtain the asymptotic behaviour of the functionals \[ F^\varepsilon(u):= \varepsilon \int \int_{I\times I} \Biggl|{u(x)- u(y)\over x- y}\Biggr|^2 dx dy+ \lambda_\varepsilon \int_I W(u) dx \] when \(\varepsilon\to 0\), \(\lambda_\varepsilon\to +\infty\) and \(\varepsilon\log \lambda_\varepsilon\to k\) \((0< k< + \infty)\).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
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