The interpolation inequality for the gradient #
The Lᵖ norm of the gradient of a function is controlled by the Lᵖ norms of the function and
of its Hessian: for every ε > 0,
‖∇u‖_p ≤ ε ‖D²u‖_p + (n² / ε) ‖u‖_p, 1 ≤ p ≤ ∞, n = dim E.
The smooth and zero-boundary estimates include p = ∞. The statements for arbitrary
whole-space Sobolev functions and between balls retain p < ∞ for the density and cutoff steps.
This is how a first-order term is absorbed into the left-hand side of a second-order estimate. In
the W^{2,p} theory of non-divergence-form operators with continuous coefficients, the
Calderón–Zygmund estimate is applied to ζ u for cutoffs ζ supported on small balls, and the
derivatives falling on ζ produce the terms ‖∇u‖_p that this inequality absorbs.
ContDiff.eLpNorm_inv_smul_comp_add_smul_sub_half_le: halving the step of a difference quotientδ_s u = s⁻¹ (u(· + s h) - u)costs at most(|s| / 4) ‖h‖² ‖D²u‖_p.ContDiff.eLpNorm_fderiv_apply_le_eLpNorm_inv_smul_comp_add_smul_sub_add: fors ≠ 0,‖Du · h‖_p ≤ ‖δ_s u‖_p + (|s| / 2) ‖h‖² ‖D²u‖_p.ContDiff.eLpNorm_fderiv_apply_le: for aC²functionuand a directionh,‖Du · h‖_p ≤ ε ‖h‖² ‖D²u‖_p + ε⁻¹ ‖u‖_p.ContDiff.eLpNorm_fderiv_le: the same for the full derivative on a finite-dimensional inner product space, with constantsεandn² / ε.TauCeti.Wkp.norm_iteratedGradient_lowerOrder_le_of_mem_wkp0Submodule: the inequality foru ∈ W^{2,p}_0(Ω), on any openΩ, with the weak gradient and the weak Hessian.TauCeti.Wkp.norm_iteratedGradient_lowerOrder_le: the inequality on all ofW^{2,p}(E).TauCeti.Wkp.exists_lpNorm_gradient_ball_le: the local form between concentric ballsB_s ⊆ B_t ⊆ Ωforu ∈ W^{2,p}(Ω), with the extra terms(t - s)⁻¹ ‖∇u‖and(t - s)⁻² ‖u‖onB_tproduced by a cutoff.TauCeti.Wkp.lpNorm_gradient_le_of_cutoff: the interpolation on an open setSfor an arbitrary smooth cutoffηthat is one onSand supported in a general openΩ, with the cutoff's gradient and Hessian boundsc₁, c₂as parameters; the Hessian of the product is estimated byTauCeti.Wkp.norm_iteratedGradient_one_le_of_value_ae_eq_mul.
The argument #
Write δ_t u = t⁻¹ (u(· + t h) - u) for the difference quotient. Halving the step changes it by a
second difference, δ_{t/2} u - δ_t u = -t⁻¹ (u(· + t h) - 2 u(· + t h / 2) + u), whose Lᵖ norm
is at most (t / 4) ‖h‖² ‖D²u‖_p by the second-order translation estimate
ContDiff.eLpNorm_comp_add_add_sub_sub_le_mul_eLpNorm_fderiv_fderiv. Starting from
‖δ_{2ε} u‖_p ≤ ε⁻¹ ‖u‖_p (MeasureTheory.eLpNorm_comp_add_sub_le_two_mul) and summing over
the dyadic steps 2ε / 2ᵏ gives ‖δ_{2ε/2ᵏ} u‖_p ≤ ε ‖h‖² ‖D²u‖_p + ε⁻¹ ‖u‖_p for every k.
The difference quotients converge pointwise to Du · h, so Fatou's lemma bounds ‖Du · h‖_p in
the same way. Summing over an orthonormal basis, ‖Du‖ ≤ ∑ᵢ |Du · eᵢ| gives the gradient bound.
For W^{2,p}_0(Ω) both sides are continuous in the Sobolev norm, and the test functions are
dense. Between concentric balls, the W^{2,p}_0 inequality is applied to the product of u with a
cutoff that is one on B_s and supported in B_t.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 7.28.
- R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2nd ed., Theorem 5.2.
Halving the step of a difference quotient. For a C² function u, a direction h and a
step s, the difference quotients δ_s u = s⁻¹ (u(· + s h) - u) satisfy
‖δ_{s/2} u‖_p ≤ ‖δ_s u‖_p + (|s| / 4) ‖h‖² ‖D²u‖_p, 1 ≤ p ≤ ∞.
A directional derivative is controlled by a single difference quotient. For a C²
function u, a direction h and a step s ≠ 0,
‖Du · h‖_p ≤ ‖δ_s u‖_p + (|s| / 2) ‖h‖² ‖D²u‖_p, 1 ≤ p ≤ ∞,
where δ_s u = s⁻¹ (u(· + s h) - u).
The interpolation inequality for a directional derivative. For a C² function u, a
direction h and ε > 0,
‖Du · h‖_p ≤ ε ‖h‖² ‖D²u‖_p + ε⁻¹ ‖u‖_p, 1 ≤ p ≤ ∞.
No support, integrability or boundedness hypothesis is needed.
The interpolation inequality for the derivative. For a C² function u on a real inner
product space E of dimension n and ε > 0,
‖Du‖_p ≤ ε ‖D²u‖_p + (n² / ε) ‖u‖_p, 1 ≤ p ≤ ∞.
No support, integrability or boundedness hypothesis is needed.
The interpolation inequality in W^{2,p}_0(Ω). Let Ω ⊆ E be open, 1 ≤ p ≤ ∞ and
ε > 0. Every u ∈ W^{2,p}_0(Ω) satisfies
‖∇u‖_p ≤ ε ‖D²u‖_p + (n² / ε) ‖u‖_p, n = dim E,
where ∇u is the weak gradient TauCeti.Wkp.iteratedGradient 0 (TauCeti.Wkp.lowerOrder 1 u) and
D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u. No boundedness or boundary
regularity of Ω is needed.
The interpolation inequality in W^{2,p}(E). Let 1 ≤ p < ∞ and ε > 0. Every
u ∈ W^{2,p}(E) satisfies
‖∇u‖_p ≤ ε ‖D²u‖_p + (n² / ε) ‖u‖_p, n = dim E,
where ∇u is the weak gradient TauCeti.Wkp.iteratedGradient 0 (TauCeti.Wkp.lowerOrder 1 u) and
D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u.
Interpolation on an open set, in terms of a cutoff. Let S be open and η a smooth
compactly supported function with tsupport η ⊆ Ω, |η| ≤ 1, equal to 1 on S, ‖∇η‖ ≤ c₁
and ‖D²η‖ ≤ c₂. Then for 1 ≤ p < ∞, u ∈ W^{2,p}(Ω) and ε > 0,
`‖∇u‖{Lᵖ(S)} ≤ ε (‖D²u‖{Lᵖ(Ω)} + 2 c₁ ‖∇u‖{Lᵖ(Ω)} + c₂ ‖u‖{Lᵖ(Ω)})
- (n² / ε) ‖u‖_{Lᵖ(Ω)}
,n = dim E`.
Interpolation between concentric balls. There is a universal constant c such that, for
1 ≤ p < ∞, every u ∈ W^{2,p}(Ω), every pair of radii 0 < s < t with B(x₀, t) ⊆ Ω, and every
ε > 0,
`‖∇u‖{Lᵖ(B_s)} ≤ ε (‖D²u‖{Lᵖ(B_t)} + c (t - s)⁻¹ ‖∇u‖{Lᵖ(B_t)} + c (t - s)⁻² ‖u‖{Lᵖ(B_t)})
- (n² / ε) ‖u‖_{Lᵖ(B_t)}
,n = dim E`.
Here B_t carries the Lᵖ(B_t) norms and the terms with c come from the derivatives of a cutoff
between the two balls; see TauCeti.Wkp.lpNorm_gradient_le_of_cutoff.