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Inverse and homogenization problems for maximal monotone operators Zarate, Ramon Saiz

Abstract

We apply self-dual variational calculus to inverse problems, optimal control problems and homogenization problems in partial differential equations. Self-dual variational calculus allows for the variational formulation of equations which do not have to be of Euler-Lagrange type. Instead, a monotonicity condition permits the construction of a self-dual Lagrangian. This Lagrangian then permits the construction of a non-negative functional whose minimum value is zero, and its minimizer is a solution to the corresponding equation. In the case of inverse and optimal control problems, we use the variational functional given by the self-dual Lagrangian as a penalization functional, which naturally possesses the ideal qualities for such a role. This allows for the application of standard variational techniques in a convex setting, as opposed to working with more complex constrained optimization problems. This extends work pioneered by Barbu and Kunisch. In the case of homogenization problems, we recover existing results by dal Maso, Piat, Murat and Tartar with the use of simpler machinery. In this context self-dual variational calculus permits one to study the asymptotic properties of the potential functional using classical Gamma-convergence techniques which are simpler to handle than the direct techniques required to study the asymptotic properties of the equation itself. The approach also allows for the seamless handling of multivalued equations. The study of such problems introduces naturally the study of the topological structures of the spaces of maximal monotone operators and their corresponding self-dual potentials. We use classical tools such as Gamma-convergence, Mosco convergence and Kuratowski-Painlevé convergence and show that these tools are well suited for the task. Results from convex analysis regarding these topologies are extended to the more general case of maximal monotone operators in a natural way. Of particular interest is that the Gamma-convergence of self-dual Lagrangians is equivalent to the Mosco convergence, and this in turn implies the Kuratowski-Painlevé convergence of their corresponding maximal monotone operators; this partially extends a classical result by Attouch relating the convergence of convex functions to the convergence of their corresponding subdifferentials.

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