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Terry Rockafellar :
Integral Functionals, Subgradients nad Duality

In many infinite-dimensional applications of nonlinear analysis, a
key role is played by functionals based on an integral expression.
This is seen in optimal control and the variational principles
behind many partial differential equations, as well as in problems
of stochastic optimization where the integral gives an expectation
with respect to a probability measure.

These lectures will cover
the basic theory of integral functionals on Lebesgue spaces and
spaces of continuous functions, with emphasis on subdifferentiation,
convexity, and conjugacy under the Legendre-Fenchel transform.