Crumples in a sheet of paper, wrinkles on curtains, cracks in metallic alloys, and defects in superconductors

are examples of patterns in materials. A thorough understanding of the underlying phenomenon behind the pattern formation provides a different prospective on the properties of the existing materials and contributes to the development of new ones. In my talk I will address the issue of modelling pattern formation via nonconvex energy minimization problems, regularized by higher order terms. Two particular examples of such models will be described in greater detail: vortices in Ginzburg-Landau model of superconductors, as well as emergence of patterns in shape-memory alloys. I will discuss the issue of well-posedness of such modelling, which in certain cases reduces to overcoming the loss of compactness in order to establish the existence of minimizers. I will also provide some examples qualitative properties of minimizers via sharp energy bounds.