Optimal transport, Euler equations, Mather and DiPerna-Lions theories

Chapter 3
Mather quotient and Sard Theorem
Let (M, g) be a smooth complete Riemannian manifold without boundary, and denote by d(x, y)
the Riemannian distance from x to y. For v ∈ TxM the norm vx is given by gx(v, v) 1/2 , and
we also denote by · x the dual norm on T ∗ M.
We assume that H : T ∗ M → R is a Hamiltonian of class C k,α , with k ≥ 2, α ∈ [0, 1], which
satisfies the three following conditions:
(H1) C 2 -strict convexity: ∀(x, p) ∈ T ∗ M, the second derivative along the fibers ∂2 H
∂p 2 (x, p) is
strictly positive definite;
(H2) uniform superlinearity: for every K ≥ 0 there exists a finite constant C(K) such that
H(x, p) ≥ Kpx + C(K), ∀ (x, p) ∈ T ∗ M;
(H3) uniform boundedness in the fibers: for every R ≥ 0, we have
sup {H(x, p) | px ≤ R} < +∞.
x∈M
By the Weak KAM Theorem it is known that, under the above conditions, there is c(H) ∈ R
such that the Hamilton-Jacobi equation
H(x, dxu) = c
admits a global viscosity solution u : M → R for c = c(H) and does not admit such solution
for c < c(H). In fact, for c < c(H), the Hamilton-Jacobi equation does not admit any viscosity
subsolution. Moreover, if M is assumed to be compact, then c(H) is the only value of c for which
the Hamilton-Jacobi equation above admits a viscosity solution. The constant c(H) is called
the critical value, or the Mañé critical value of H. In the sequel, a viscosity solution u : M → R
of H(x, dxu) = c(H) will be called a critical viscosity solution or a weak KAM solution, while
a viscosity subsolution u of H(x, dxu) = c(H) will be called a critical viscosity subsolution (or
critical subsolution if u is at least C 1 ).
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