Decay estimates for harmonic functions on concentric balls #
Let E be a finite-dimensional real inner product space of dimension n, with an additive Haar
measure μ, and let u : E → F be harmonic on the ball B_R = ball x₀ R. This file proves the
two decay estimates on concentric balls B_ρ ⊆ B_R that a harmonic function satisfies:
∫_{B_ρ} ‖u‖² ≤ 2ⁿ (ρ / R)ⁿ ∫_{B_R} ‖u‖²,
∫_{B_ρ} ‖u - u_ρ‖² ≤ 2^(3n + 8) (n + 1)² (ρ / R)^(n + 2) ∫_{B_R} ‖u - c‖²
for every 0 ≤ ρ ≤ R and every constant c, where u_ρ is the average of u over B_ρ. The
constants depend only on the dimension. The first estimate says that the L² mass of u on
small balls decays like the volume, as it would for a bounded function; the second says that the
L² mean oscillation decays two orders faster, as it would for a Lipschitz function.
These are the decay estimates of the frozen problem in the Campanato approach to elliptic
regularity, when the frozen operator is the Laplacian. There a solution of a perturbed equation,
such as Poisson's equation or an equation with variable coefficients, is compared on each ball
with the harmonic function having the same boundary values. The decay exponents n and n + 2
feed the iteration lemma TauCeti.exists_le_mul_div_rpow_mul_add_mul_rpow_of_le_mul_add, whose
output is a growth bound that Campanato's characterization of Hölder continuity
(TauCeti.MeasureTheory.exists_holderWith_ae_eq_iff_forall_setAverage_norm_sub_le) turns into a
Hölder estimate. The function u is only assumed square integrable on the open ball B_R, as for
a harmonic function with prescribed H¹ boundary values.
The argument #
Everything rests on the mean-value property on balls
(InnerProductSpace.HarmonicOnNhd.setIntegral_ball_eq). Combined with the Cauchy–Schwarz
inequality it gives the pointwise bound μ(B(y, s)) ‖u y‖² ≤ ∫_{B(y, s)} ‖u‖², so ‖u‖² is bounded
on B_{R/2} by 2ⁿ / μ(B_R) times ∫_{B_R} ‖u‖²; integrating this bound over B_ρ gives the
first estimate.
For the second, comparing the mean values of u over two balls B(y, s) and B(z, s) of the
same radius leaves only the integrals of u over the two differences, each contained in a
spherical shell of width ‖y - z‖ (the comparison behind Nelson's proof of Liouville's theorem).
If ‖u‖ ≤ M on both balls this gives the Lipschitz bound
‖u y - u z‖ ≤ 2ⁿ n M ‖y - z‖ / s for ‖y - z‖ ≤ s, the difference form of the interior gradient
estimate. Applied to u - c, with M from the pointwise bound, it controls the oscillation of
u on B_ρ by (ρ / R) times the L² norm of u - c on B_R, for ρ ≤ R / 8. For larger ρ
the estimate follows from replacing the average u_ρ by c
(TauCeti.setIntegral_norm_sub_setAverage_sq_le).
Linear changes of variables #
Both estimates survive a linear change of variables, with constants depending on its condition
number. If e : E' ≃L[ℝ] E is a continuous linear equivalence with ‖e⁻¹ x‖ ≤ a ‖x‖ and
‖e y‖ ≤ b ‖y‖, and u ∘ e is harmonic on e⁻¹(B_R), then u satisfies the two estimates with
the factors (a b)ⁿ and 4 (a b)^(n + 2) inserted. The ball B_ρ is pulled back into a ball of
radius a ρ and a ball of radius R / b is pushed into B_R, and the harmonic estimates are
applied for the Haar measure μ.map e⁻¹, for which e carries one integral to the other without
a Jacobian factor. A second-order operator with constant elliptic coefficients becomes the
Laplacian after such a change of variables, so this is the decay estimate for its solutions.
Main declarations #
InnerProductSpace.HarmonicOnNhd.measureReal_ball_mul_norm_sq_le_setIntegral: the pointwise boundμ(B(y, s)) ‖u y‖² ≤ ∫_{B(y, s)} ‖u‖².InnerProductSpace.HarmonicOnNhd.setIntegral_ball_norm_sq_le: decay of theL²mass with exponentn.InnerProductSpace.HarmonicOnNhd.norm_sub_le_of_norm_le: the Lipschitz bound for a bounded harmonic function.InnerProductSpace.HarmonicOnNhd.setIntegral_ball_norm_sub_setAverage_sq_le: decay of theL²mean oscillation with exponentn + 2.TauCeti.setIntegral_ball_norm_sq_le_of_harmonicOnNhd_compandTauCeti.setIntegral_ball_norm_sub_setAverage_sq_le_of_harmonicOnNhd_comp: the two estimates foruwhenu ∘ eis harmonic,ea continuous linear equivalence.InnerProductSpace.HarmonicOnNhd.setIntegral_ball_norm_sub_setAverage_sq_le_add: the comparison step of the Campanato approach. A square-integrablefclose inL²(B_R)to a harmonic function inherits its decay, up to the distance∫_{B_R} ‖f - u‖².
References #
- M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Chapter 5 (decay estimates for constant coefficients and the Campanato approach to Schauder estimates).
- M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Chapter III.
- E. Nelson, A proof of Liouville's theorem, Proc. Amer. Math. Soc. 12 (1961), 995.
- L. C. Evans, Partial Differential Equations, Section 2.2.3, Theorem 7 (estimates on derivatives).
The pointwise L² bound for harmonic functions. If u is harmonic on a neighbourhood of
the closed ball closedBall y s, then the measure of the ball ball y s times ‖u y‖² is at most
the integral of ‖u‖² over that ball.
Decay of the L² mass of a harmonic function. If u is harmonic on the ball
ball x₀ R and ‖u‖² is integrable there, then for every 0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖u‖² ≤ 2ⁿ (ρ / R)ⁿ ∫_{ball x₀ R} ‖u‖²,
where n is the dimension of E.
The Lipschitz bound for bounded harmonic functions. Let u be harmonic on neighbourhoods
of the closed balls closedBall y s and closedBall z s, with 0 < s and dist y z ≤ s, and let
‖u‖ ≤ M on the open balls ball y s and ball z s. Then
‖u y - u z‖ ≤ 2ⁿ n M dist y z / s,
where n is the dimension of E. This is the difference form of the interior gradient estimate
|∇u| ≤ C M / s for harmonic functions.
The actual Fréchet derivative of a harmonic function bounded by M on ball x (2 * ρ)
has norm at most 2ⁿ n M / ρ, where n is the domain dimension.
Decay of the L² mean oscillation of a harmonic function. If u is harmonic on the ball
ball x₀ R and ‖u - c‖² is integrable there for some constant c, then for every 0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖u - u_ρ‖² ≤ 2^(3n + 8) (n + 1)² (ρ / R)^(n + 2) ∫_{ball x₀ R} ‖u - c‖²,
where u_ρ is the average of u over ball x₀ ρ and n is the dimension of E. Taking for c
the average of u over ball x₀ R gives the Campanato decay estimate of the mean oscillation.
Linear changes of variables #
Decay of the L² mass after a linear change of variables. Let e : E' ≃L[ℝ] E be a
continuous linear equivalence with ‖e.symm x‖ ≤ a ‖x‖ and ‖e y‖ ≤ b ‖y‖, and let u ∘ e
be harmonic on e ⁻¹' ball x₀ R, with ‖u‖² integrable on ball x₀ R. Then for every 0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖u‖² ≤ 2ⁿ (a b)ⁿ (ρ / R)ⁿ ∫_{ball x₀ R} ‖u‖²,
where n is the dimension of E. The constant depends on e only through the bound a b for
its condition number.
Decay of the L² mean oscillation after a linear change of variables. Let
e : E' ≃L[ℝ] E be a continuous linear equivalence with ‖e.symm x‖ ≤ a ‖x‖ and
‖e y‖ ≤ b ‖y‖, and let u ∘ e be harmonic on e ⁻¹' ball x₀ R, with ‖u - c‖² integrable on
ball x₀ R for some constant c. Then for every 0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖u - u_ρ‖² ≤ C (a b)^(n + 2) (ρ / R)^(n + 2) ∫_{ball x₀ R} ‖u - c‖²
with C = 2^(3n + 10) (n + 1)²,
where u_ρ is the average of u over ball x₀ ρ and n is the dimension of E.
Comparison with a harmonic function. Let H be harmonic on B_r = ball x₀ r in a space
of dimension n, and let f and H be square integrable on B_r. For 0 < ρ ≤ r, the square
mean oscillation ψ(t) = ∫_{B_t} ‖f - f_{B_t}‖² of f decays like that of H, up to the
distance ∫_{B_r} ‖f - H‖² between them:
ψ(ρ) ≤ 4D (ρ / r)^(n + 2) ψ(r) + (4D + 2) ∫_{B_r} ‖f - H‖², where D = 2^(3n + 8) (n + 1)² is
the constant of InnerProductSpace.HarmonicOnNhd.setIntegral_ball_norm_sub_setAverage_sq_le.