Trudinger's inequality on a closed Riemannian manifold #
Let M be a compact manifold without boundary, of dimension n ≥ 2, with a continuous
Riemannian metric and its Riemannian volume dV. Trudinger's inequality says that a function
of unit W^{1,n} norm is exponentially integrable: there are constants β > 0 and C such that
∫_M exp (β |u| ^ (n / (n - 1))) dV ≤ C
for every C¹ function u with ‖∇u‖_{Lⁿ} + ‖u‖_{Lⁿ} ≤ 1
(TauCeti.Manifold.exists_lintegral_exp_mul_abs_rpow_le). On a closed surface, n = 2, this is
the two-dimensional exponential estimate ∫_M exp (β u²) dV ≤ C for ‖∇u‖_{L²} + ‖u‖_{L²} ≤ 1
(TauCeti.Manifold.exists_lintegral_exp_mul_sq_le_of_finrank_eq_two). It is what makes e^(2u)
integrable, with a bound in terms of the H¹ norm of u, in the variational treatment of the
prescribed Gaussian curvature equation Δ⁺u + K₀ = K e^(2u) on a closed surface.
The proof localizes the flat inequality
TauCeti.setLIntegral_exp_norm_div_rpow_le_of_eLpNorm_fderiv_le. Near each point c, in a
coordinate ball around the chart image of c, the chart density is comparable to its value at c
and the coordinate derivative of every function is bounded by a fixed multiple of the length of
its gradient
(TauCeti.Manifold.eventually_norm_fderiv_comp_extChartAt_symm_le_mul_norm_mgradient). A bump
function equal to 1 on a smaller coordinate ball cuts u off to a compactly supported C¹
function on the model space, whose derivative has Lⁿ norm bounded by a constant depending only
on c, the metric and the bump; the flat inequality then bounds the integral of
exp (β |u| ^ (n / (n - 1))) over the chart preimage of the smaller ball
(TauCeti.Manifold.exists_mem_nhds_setLIntegral_exp_mul_abs_rpow_le). Finitely many such
neighbourhoods cover M.
Main results #
TauCeti.Manifold.exists_mem_nhds_setLIntegral_exp_mul_abs_rpow_le: the local inequality, near any point of a manifold without boundary.TauCeti.Manifold.exists_lintegral_exp_mul_abs_rpow_le: Trudinger's inequality on a closed Riemannian manifold of dimension at least two.TauCeti.Manifold.exists_lintegral_exp_mul_sq_le_of_finrank_eq_two: its form on a closed surface.
References #
- N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.
- T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer (1998), Chapter 2.
- J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974), 14–47.
Trudinger's inequality #
Trudinger's inequality near a point. Every point c of a manifold without boundary, of
dimension n ≥ 2, with a continuous Riemannian metric, has a neighbourhood V with constants
β > 0 and C such that ∫_V exp (β |u| ^ (n / (n - 1))) dV ≤ C for every C¹ function u
with ‖∇u‖_{Lⁿ} + ‖u‖_{Lⁿ} ≤ 1.
Trudinger's inequality on a closed Riemannian manifold. On a compact manifold without
boundary, of dimension n ≥ 2, with a continuous Riemannian metric, there are constants β > 0
and C such that ∫_M exp (β |u| ^ (n / (n - 1))) dV ≤ C for every C¹ function u with
‖∇u‖_{Lⁿ} + ‖u‖_{Lⁿ} ≤ 1.
The two-dimensional exponential estimate. On a closed surface with a continuous Riemannian
metric, there are constants β > 0 and C such that ∫_M exp (β u²) dV ≤ C for every C¹
function u with ‖∇u‖_{L²} + ‖u‖_{L²} ≤ 1.