1D symmetry for semilinear PDEs from the limit interface of the solution

Alberto Farina, Enrico Valdinoci

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4 Citations (Scopus)

Abstract

We study bounded, monotone solutions of Δu = W′(u) in the whole of ℝn, where W is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, u is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of Γ-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and wishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.
Original languageEnglish
Pages (from-to)665-682
Number of pages18
JournalCommunications in Partial Differential Equations
Volume41
Issue number4
DOIs
Publication statusPublished - 2016
Externally publishedYes

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