## The gradient theory of phase transitions for systems with two potential wells.(English)Zbl 0676.49005

Summary: We generalise the gradient theory of phase transitions to the vector valued case. We consider the family of perturbations $E_{\epsilon}(u):=\int_{\Omega}W(u)dx\quad +\quad \epsilon^ 2\int_{\Omega}| \nabla u|^ 2 dx$ of the nonconvex functional $$E_ 0(u):=\int_{\Omega}W(u)dx$$, where W: $${\mathbb{R}}^ N\to {\mathbb{R}}$$ supports two phases and $$N\geq 1$$. We obtain the $$\Gamma$$ (L$${}^ 1(\Omega))$$-limit of the sequence $$J_{\epsilon}(u):=\epsilon^{-1}E_{\epsilon}(u)$$. Moreover, we improve a compactness result ensuring the existence of a subsequence of minimisers of $$E_{\epsilon}(\cdot)$$ converging in $$L^ 1(\Omega)$$ to a minimiser of $$E_ 0(\cdot)$$ with minimal interfacial area.

### MSC:

 49J99 Existence theories in calculus of variations and optimal control 49J52 Nonsmooth analysis
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### References:

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