Singular kernels, multiscale decomposition of microstructure, and dislocation models (Q717448)

The paper studies a model functional arising in the study of phase field models for dislocations and containing two terms, namely: a non-convex term which favors integer values of the vector-valued phase field \(u\), and a regularization term. The functional regularization via the Dirichlet integral is replaced by a singular non-local term which behaves as the \(H^{1/2}\) norm. The logarithmic failure of the embedding of \(H^{1/2}\) into continuous function reflects the fact that all length scales play a role, and the appropriate rescaling is logarithmic. This eliminates the possibility of selecting one dominant length scale and focusing on a cell problem on that scale. Moreover, the functional is vectorial and anisotropic, and the lower-order term has infinitely many minima. In the scalar (isotropic) case, the reduced functional has been studied by \textit{G. Alberti, G. Bouchitté} and \textit{P. Seppecher} [C. R. Acad. Sci., Paris, Sér. I 319, No. 4, 333--338 (1994; Zbl 0845.49008)] which proved a compactness result, i.e., the domain of the limiting functional is \(BV(\Omega; {W=0})\), where in general case \(W\) is a multiwell potential and \(\Omega\) belongs to \(R^n\). The reformulation of the phase field model has been proposed by \textit{M. Koslowski, A. M. Cuitiño} and \textit{M. Ortiz} [J. Mech. Phys. Solids 50, No. 12, 2597--2635 (2002; Zbl 1094.74563)] (KCO) for dislocations on a given slip plane. Considering the KCO model, the authors obtain a sharp-interface limit of the model within the framework of \(\Gamma\)-convergence. It is used the fact the \(BV\)-function cannot have significant microstructure on all scales simultaneously, and the few potentially bad scales can be ignored in the limit. For the \(BV\)-relaxation of 1D interfacial energy, the authors formulate the main theorem. First, some elementary results for non-local terms with integrable kernels are considered. Then, the singular kernel is decomposed into a sequence of integrable kernels and it is shown that the regular phase field can be replaced with a \(BV\)-function with values in \(Z^n\). Further, it is shown that if a function is 1D and takes values in \(Z^n\), then it is possible to estimate its non-local truncated energy, with the right line tension energy. Due to this, the functions with well-controlled energy can be approximated by 1D functions. Then, by combining the last two results, the authors obtain a global approximation connected with the construction of a new function such that the relaxed line energy is essentially controlled by the truncated energy of \(u\). To quantify the distance of the \(BV\)-function from a locally 1D function on a given length scale, the authors use an iterative mollification on different length scales, starting from the smallest one, and measure the defect in the total variation of the gradient. Finally, the authors prove the lower and upper bounds for the main theorem.