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Abstract: The classical Allen-Cahn equation gives a bridge between the

theory of phase transition and the theory of minimal surfaces. In this talk

I will discuss the existence of multiple solutions for a suitable variant

of this equation, satisfying a volume constraint. This aims naturally at an

existence theory for constant mean curvature hypersurfaces. Joint work

with Vieri Benci (Pisa) and Stefano Nardulli (UFABC).