Parametrized Measures and Variational Principles

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Weak convergence is essential in modern nonlinear analysis due to its compactness properties similar to finite-dimensional spaces, where bounded sequences correspond to weakly relatively compact sets. However, weak convergence presents challenges with nonlinear functionals and operations, complicating nonlinear analysis beyond expectations. Parametrized measures help to understand weak convergence and its interaction with nonlinear functionals, providing a means to represent weak limits of compositions with nonlinear functions through integrals. This approach is particularly useful for analyzing oscillatory phenomena and tracking changes in oscillations when nonlinear functionals are applied. Additionally, weak convergence is crucial in the calculus of variations, as uniform bounds in norm for sequences enable the existence of weakly convergent subsequences. Establishing weak lower semicontinuity is vital for demonstrating the existence of minimizers for specific functionals, representing a key step in the direct method of the calculus of variations. Significant research has focused on identifying conditions under which weak lower semicontinuity holds for nonlinear functionals expressed as integrals. The findings indicate that some form of convexity, interpreted broadly, is typically involved in these scenarios.