Albert Fathi (Georgia Institute of Technology, USA)
Viscosity solutions of the Hamilton-Jacobi equations
on non-compact manifolds

(10 hours)

- Lecture 1: Tuesday June 1, 11-13  (UTC+2)
- Lecture 2: Friday June 4, 11-13  (UTC+2)
- Lecture 3: Tuesday June 15, 11-13  (UTC+2)
- Lecture 4: Thursday June 17, 11-13  (UTC+2)
- Lecture 5: Tuesday June 22, 11-13  (UTC+2)

Lectures will be streamed by the platform Microsoft Teams.
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The purpose of this course is to study the properties of viscosity solutions of the Hamilton-Jacobi equations on non-compact manifolds, in the spirit of what was done for the case of compact manifolds in

Albert Fathi
, Weak KAM from a PDE point of view: viscosity solutions of the Hamilton- Jacobi equation and Aubry set, Proc. Roy. Soc. Edinburgh Sect. A, 120 (2012) 1193–1236

We will be mainly interested in viscosity solutions of the evolution Hamilton-Jacobi equation

t U + H (x, ∂x U ) = 0.

Here we think of the case where U : [0, +[×M R, with M is a manifold.
M is compact, as has been known for a long time, the maximum principle yields uniqueness for a given initial condition U|{0M. This in turn implies the representation
by a Lax-Oleinik type formula.

When M is not compact, the global maximum principle does not immediately hold.

We will show how to obtain the Lax-Oleinik formula and the uniqueness result. We will consider the pointwise finiteness of the Lax-Oleinik formula for general initial conditions. results.

We will also discuss results on the topology of the set of singularities of such solutions and give applications to Riemannian Geometry.

Note: This series of lectures is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006