Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order problems and to models involving lower dimension density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and the calculus of variations.

In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others higher order terms play a fundamental role.

Applications to quantum dots in epitaxy deposition, and recolorization and denoising in imaging science will be addressed.