Variational principles and their applications in modern analysis

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Variational methods provide efficient and elegant tools for formulating and solving mathematical problems that are of key importance to scientists and engineers, and their applications span a wide range that includes optimization theory, non-smooth analysis, economics, control theory, as well as image processing. This is not surprising, since many problems in science and engineering can be recast as variational problems. However, it is known that not every equation admits the standard Euler-Lagrange variational structure. In recent years, considerable effort has been devoted to the development of a calculus of variations that would go beyond this standard variational structure and that could be applied to a large number of partial differential systems and evolution equations, many of which do not fit within the classical Euler-Lagrange framework. The birth of the theory of convex and non-convex self-duality has been the result of some of these studies. The Weighted energy-dissipation functionals is another novel approach in this direction. My aim is to extend further these variatioal principles to deal with a larger class of equations beyond the classical range. I would also like to investigate their applications to inverse problems, Monge's mass transport and problems in elasticity.******