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Uniform convergence of a singular perturbation problem. (English) Zbl 0829.49013
Let \(J_\varepsilon (u, \Omega)\) be the functional defined by \[ J_\varepsilon (u, \Omega)= \int_\Omega \bigl\{ \varepsilon|\nabla u|^2+ F(x,u) \bigr\}dx, \] \(\varepsilon>0\), \(\Omega\) bounded open domain, \(F\geq 0\), \(u\in \{w\in H^1(\Omega)\): \(u^1\leq w\leq u^2\}\), \(F(x, u^i)=0\).
In this paper the authors answer to the question of how a sequence of minimizers \(\{u_\varepsilon\}\) converges to \(u_0\) by proving the uniform convergence of the level sets of \(u_\varepsilon\) to the limiting surface.

49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI
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