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.

1D symmetry for semilinear PDEs from the limit interface of the solution / A. Farina, E. Valdinoci. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - 41:4(2016), pp. 665-682.

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

E. Valdinoci
2016

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.
Limit interface; phase transitions; symmetry results
Settore MAT/05 - Analisi Matematica
Elliptic Pdes and Symmetry of Interrfaces and Layers for Odd Nonlinearties
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/472831
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