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Stability of invariant manifolds in one and two dimensions
Giovanni Bellettini, Anna De Masi, Nicolas Dirr and Errico Presutti
We consider the gradient flow associated with a non local free energy functional and extend to such a case results obtained for the Allen-Cahn equation on "slow motions on invariant manifolds". The manifolds in question are time-invariant one-dimensional curves in a $L^2$ space which connect the two ground states (interpreted as the pure phases of the system) to the first excited state (interpreted as a diffuse interface). Local and structural stability of the manifolds are proved and applications to the characterization of optimal tunnelling are discussed.