Higher Order Variational Problems

Higher order variational problems appear often in the engineering literature and in connection with the so-called gradient theories of phase transitions within elasto-plastic regimes. The study of equilibria of micromagnetic materials asks for mastery of second order energies (see [51], [91]; see also [31], [38], [44], [45], [61], [77], [78], [79], [108]), and the Blake-Zisserman model for image segmentation in computer vision (see [34], [35], [36]; see also [50]) seats squarely among second-order free discontinuity models which may be recasted as higher order Griffiths’ models for fracture mechanics (see [7], [24], [28],[58], [68], [69], [70], [71]). The energy functionals may include lower dimensional order terms to take into account interfacial energies and discontinuities of underlying fields (see [10]). Here we will neglect the role played by these terms and we will focus on the added difficulties inherent to the presence of derivatives of order two or more.