Abstract
We first present some examples of Hamilton-Jacobi equations in infinite dimensions, we solve formally these equations and we see that in all examples presented, “solutions” are not smooth. We also present minimization problems without compactness. Variational principles are efficient tools to deal with these minimization problems. We then show how the smooth variational principle of R. Deville, G. Godefroy and V. Zizler allows to develop a differential calculus for non-smooth functions in smooth Banach spaces. This calculus is then applied to the resolution of Hamilton-Jacobi equations in infinite dimensions: we prove that in smooth Banach spaces, this calculus yields uniqueness results of viscosity solutions of Hamilton-Jacobi equations. We also present the key facts for the resolution of second order fully non-linear partial differential equations. Several open problems connected with the content of these lectures will also be presented.
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Deville, R. (1999). Smooth variational principles and non-smooth analysis in Banach spaces. In: Clarke, F.H., Stern, R.J., Sabidussi, G. (eds) Nonlinear Analysis, Differential Equations and Control. NATO Science Series, vol 528. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4560-2_6
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DOI: https://doi.org/10.1007/978-94-011-4560-2_6
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