Local minimizers induced by spatial inhomogeneity with inner transition layer
This paper is concerned with the existence and the geometric structure of nonconstant minimizers in \(L_1(\Omega)\) of singularly perturbed nonconvex variational problems of the type \[ F_\epsilon(v) = \int_\Omega [\epsilon^2 k_1^2(x)|\nabla v|^2 + k_2^2(x)(v^2-1)^2] dx, \] where \(k_1\) and \(k_2\) are positive regular functions. For each closed simple smooth curve \(\gamma\) contained in a level set of \(k_1(x) \cdot k_2(x), \;x \in \Omega\), necessary conditions are obtained for the variational problem to possess a nonconstant minimizer \(v_\epsilon\). This part can be applied to any two-dimensional domain \(\Omega\) and generalizes previous results when \(\Omega\) is the unit ball. Moreover, \(v_\epsilon \rightarrow 1,\) as \(\epsilon \rightarrow 0,\) uniformly on compact sets contained in one of the two connected components of \(\Omega \setminus \gamma\) and \(v_\epsilon \rightarrow -1,\) as \(\epsilon \rightarrow 0,\) uniformly on compact sets contained in the other one.
singularly perturbed nonconvex variational problems
scientific article; zbMATH DE number 1000151