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

Abstract : We study bounded, monotone solutions of u=W(u) in the whole of (n), 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 -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.
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https://hal-u-picardie.archives-ouvertes.fr/hal-03621423
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Soumis le : lundi 28 mars 2022 - 10:57:11
Dernière modification le : mardi 29 mars 2022 - 03:58:27

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Alberto Farina, Enrico Valdinoci. 1D symmetry for semilinear PDEs from the limit interface of the solution. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2016, 41 (4), pp.665-682. ⟨10.1080/03605302.2015.1135165⟩. ⟨hal-03621423⟩

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