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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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