Convex functions of supersolutions and the lower bound for positive supersolutions #
Let a be measurable and uniformly elliptic on Ω ⊆ ℝⁿ with constants 0 < λ ≤ Λ, and let
u ∈ H¹(Ω) be a weak supersolution of the divergence-form equation
-∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0 in Ω,
meaning a(u, v) ≥ 0 for every nonnegative v ∈ H¹₀(Ω). If u ≥ m almost everywhere and F is
convex and nonincreasing on [m, ∞), then F(u) is a weak subsolution:
a(F(u), v) ≤ 0 for every nonnegative v ∈ H¹₀(Ω). The weak gradient of F(u) is
F'(u) ∇u, and the formal computation is
a(F(u), v) = ∫ F'(u) ⟨a ∇u, ∇v⟩ = -a(u, -F'(u) v) - ∫ F''(u) v ⟨a ∇u, ∇u⟩ ≤ 0,
using -F'(u) v ≥ 0 as a test function for the supersolution u, and F'' ≥ 0. This is only
formal because -F'(u) v need not lie in H¹₀(Ω) for a general v; the proof carries it out
for v a test function and extends the inequality by density (see the implementation notes).
The main application is to the reciprocal t ↦ 1/t of a supersolution bounded below by a
positive constant. The reciprocal itself is not C² and does not vanish at 0, so it is
approached by the functions t ↦ t / (t² + c²): these are convex and nonincreasing on
[2c, ∞), so each u / (u² + c²) is a subsolution once 2c ≤ m, and De Giorgi's local
boundedness theorem for subsolutions bounds them from above, uniformly in c. Letting c → 0
bounds u⁻¹. More generally, a local Lᵖ upper bound for subsolutions transfers to a bound
for u⁻¹ in terms of its Lᵖ norm, for every p > 0, without dependence on the lower bound m.
This supplies the negative-power estimates in Moser's weak Harnack argument.
Under a Sobolev inequality ‖v‖_q ≤ S ‖∇v‖₂, with α = 1 - 2/q > 0, the L² form is
u⁻¹ ≤ D S^{1/α} R^{-1/α} ‖u⁻¹‖_{L²(B(x₀, R))} on B(x₀, R/2).
The scale-invariant estimates from small negative powers are in
TauCeti.Analysis.PDE.Regularity.Supersolution.Lp.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.energyFormH1_contDiffComp_nonpos: a convex nonincreasing function of a supersolution is a subsolution.TauCeti.PDE.exists_ae_inv_value_le_mul_rpow_mul_rpow_mul_sqrt_setIntegral_of_forall_eLpNorm_le: the lower boundu⁻¹ ≤ D S^{1/α} R^{-1/α} ‖u⁻¹‖_{L²(B(x₀, R))}for positive supersolutions, under a Sobolev inequality‖v‖_q ≤ S ‖∇v‖₂onW^{1,2}_0(B(x₀, R))withq > 2andα = 1 - 2/q.TauCeti.PDE.UniformlyEllipticOn.ae_inv_value_le_mul_setIntegral_rpow: transfer of a local subsolution upper bound to a reciprocal supersolution bound at every positive exponent.
Implementation notes #
The test function -F'(u) v is a product of two Sobolev functions, which need not lie in
H¹₀(Ω) for a general v. It is first formed for a test function φ ∈ C_c^∞(Ω) as the positive
part of -φ F'(u), and the inequality a(F(u), φ⁺) ≤ 0 obtained for it passes to every
v ∈ H¹₀(Ω) in the form a(F(u), v⁺) ≤ 0, by density of the test functions and continuity of
v ↦ v⁺ on H¹(Ω).
The reciprocal is approached through the smooth functions t ↦ t / (t² + c²), which vanish at
0, have bounded first and second derivatives, are convex and decreasing on [√3 c, ∞), and
increase to 1/t as c → 0.
References #
- J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Chapter 8.
- Q. Han, F. Lin, Elliptic Partial Differential Equations, Chapter 4.
Convex nonincreasing functions of supersolutions #
A convex nonincreasing function of a supersolution is a subsolution. Let a be measurable
and uniformly elliptic on Ω, and let u ∈ H¹(Ω) be a weak supersolution of
-∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0, that is a(u, v) ≥ 0 for every nonnegative v ∈ H¹₀(Ω), with u ≥ m
almost everywhere on Ω. Let F be C² with bounded first and second derivatives and
F(0) = 0, so that F(u) ∈ H¹(Ω) (TauCeti.W1p.contDiffComp), and suppose that F' ≤ 0 and
F'' ≥ 0 on [m, ∞). Then F(u) is a weak subsolution: a(F(u), v) ≤ 0 for every
nonnegative v ∈ H¹₀(Ω).
The condition at 0 and the global bounds on F' and F'' only make F(u) and the localized
products φ F'(u), for test functions φ ∈ C_c^∞(Ω), Sobolev functions; the sign conditions are
used on [m, ∞) alone.
The lower bound for positive supersolutions #
Transfer a local Lᵖ upper bound for weak subsolutions to a lower bound for positive
weak supersolutions. If every subsolution w satisfies
w ≤ K (∫ ((w)⁺)ᵖ)^{1/p} on the half ball, then
u⁻¹ ≤ K (∫ u⁻ᵖ)^{1/p} there, for every p > 0.
The constant is unchanged and does not depend on the positive lower bound for u.
The lower bound for positive supersolutions under a Sobolev inequality. Fix ellipticity
constants λ, Λ and an exponent q > 2, and let α = 1 - 2/q. There is D > 0, depending only
on these (and the dimension), such that the following holds. Let a be measurable and uniformly
elliptic on Ω with constants λ, Λ, and let u ∈ H¹(Ω) be a weak supersolution of
-∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0, that is a(u, v) ≥ 0 for every nonnegative v ∈ H¹₀(Ω), with u ≥ m > 0
almost everywhere on Ω. If B(x₀, R) ⊆ Ω and ‖v‖_q ≤ S ‖∇v‖₂ for every
v ∈ W^{1,2}_0(B(x₀, R)), then
u⁻¹ ≤ D S^{1/α} R^{-1/α} (∫_{B(x₀, R)} u⁻²)^{1/2} almost everywhere on B(x₀, R/2).
The constant does not depend on m or on S.