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This book presents three short courses on topics at the intersection of Calculus of Variations, PDEs and Material Science, based on lectures given at the CIME summer school “Variational and PDE Methods in Nonlinear Science”, held in Cetraro (Italy), July 10–14, 2023. Fabrice Bethuel discusses aympototics for Allen–Cahn systems, providing an overview of classical methods and tools for the scalar case and further results for the two-dimensional vectorial case. An alternate monotonicity formula is described, and the still open parabolic vectorial case is considered. Angkana Rüland...

This book is concerned with the study in two dimensions of stationary solutions of u_ɛ of a complex valued Ginzburg-Landau equation involving a small parameter ɛ. Such problems are related to questions occurring in physics, e.g., phase transition phenomena in superconductors and superfluids. The parameter ɛ has a dimension of a length which is usually small. Thus, it is of great interest to study the asymptotics as ɛ tends to zero.

One of the main results asserts that the limit u-star of minimizers u_ɛ exists. Moreover, u-star is smooth...

The original motivation of this study comes from the following questions that were mentioned to one ofus by H. Matano. Let 2 2 G= B = {x=(X1lX2) E 2; x + x = Ixl