Harmonic replacement #
Let u ∈ H¹(Ω). Its harmonic replacement on Ω is the weak solution w of the Dirichlet
problem Δw = 0 in Ω, w = u on ∂Ω: the function w ∈ H¹(Ω) with u - w ∈ H¹₀(Ω) and
∫_Ω ⟨∇w, ∇v⟩ = 0 for every v ∈ H¹₀(Ω). Replacing a function by the solution of the frozen
problem with the same boundary values on each ball is the comparison step of Campanato's approach
to Schauder estimates. Two facts make it work:
- the replacement is harmonic (after modification on a null set), so the decay estimates of harmonic functions apply to it;
- the error
u - wis controlled in energy by the equationusatisfies. If∫_Ω ⟨∇u, ∇v⟩ = ⟪G, ∇v⟫_{L²}for everyv ∈ H¹₀(Ω), for instance whenusolves-Δu = -div Gweakly, then‖∇(u - w)‖_{L²(Ω)} ≤ ‖G‖_{L²(Ω)}.
The energy estimate holds for every measurable, uniformly elliptic a with constants
0 < λ ≤ Λ, in the form λ ‖∇(u - w)‖ ≤ ‖G‖ for any w with u - w ∈ H¹₀(Ω) and
a(w, v) = 0 for every v ∈ H¹₀(Ω), as is needed when the frozen problem has a constant
coefficient matrix other than the identity. Its proof is one line: test the equation for
u - w ∈ H¹₀(Ω) against itself and use ellipticity.
Harmonicity comes from Weyl's lemma. A function w ∈ H¹(Ω) with ∫_Ω ⟨∇w, ∇φ⟩ = 0 for every
test function φ on Ω is weakly harmonic, ∫ Δφ • w = -∫ ⟪∇φ, ∇w⟫ = 0, so it agrees almost
everywhere on Ω with a harmonic function. Existence and uniqueness of the replacement on a
bounded domain are the Dirichlet problem with boundary values in H¹(Ω),
UniformlyEllipticOn.existsUnique_sub_mem_w1p0Submodule_forall_energyFormH1_eq_of_subset_ball.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.mul_norm_gradient_sub_le: the energy estimateλ ‖∇(u - w)‖ ≤ ‖G‖for the replacement relative to a general elliptica.TauCeti.PDE.exists_harmonicOnNhd_ae_eq_of_forall_energyFormH1_eq_zero: a function ofH¹(Ω)that solvesΔw = 0weakly agrees almost everywhere with a harmonic function.TauCeti.PDE.W1p.exists_harmonicOnNhd_ae_eq_gradient: so does its weak gradient, since the weak derivatives of a weakly harmonic function are weakly harmonic.TauCeti.PDE.exists_sub_mem_w1p0Submodule_harmonicOnNhd_ae_eq_of_subset_ball: harmonic replacement on a bounded domain, with its harmonic representative and the energy estimate.
References #
- M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd ed., Section 5.3 (Schauder estimates by comparison with the frozen problem).
- L. C. Evans, Partial Differential Equations, 2nd ed., Section 6.2.
The energy estimate for the replacement. Let a be measurable and uniformly elliptic on
Ω with constants 0 < λ ≤ Λ, and let u, w ∈ H¹(Ω) have the same boundary values,
u - w ∈ H¹₀(Ω), with w a weak solution of -∂ᵢ(aⁱʲ ∂ⱼw) = 0, that is a(w, v) = 0 for
every v ∈ H¹₀(Ω). If the functional v ↦ a(u, v) on H¹₀(Ω) is represented by a field
G ∈ L²(Ω; ℝⁿ) as a(u, v) = ⟪G, ∇v⟫_{L²}, then λ ‖∇(u - w)‖_{L²} ≤ ‖G‖_{L²}.
Weakly harmonic Sobolev functions are harmonic. If w ∈ H¹(Ω) solves Δw = 0 weakly,
in the sense that ∫_Ω ⟨∇w, ∇φ⟩ = 0 for every test function φ on Ω, then w agrees almost
everywhere on Ω with a function harmonic on Ω. No boundary condition on w and no regularity
of Ω is assumed.
The weak gradient of a weakly harmonic Sobolev function is harmonic. If the value of
w ∈ W^{1,p}(Ω) is weakly harmonic on Ω, then the weak gradient of w agrees almost
everywhere on Ω with a function Ω → ℝⁿ harmonic on Ω.
Harmonic replacement. Let Ω ⊆ ℝ^{n+1} be an open set contained in a ball, and let
u ∈ H¹(Ω). There is w ∈ H¹(Ω) with the same boundary values, u - w ∈ H¹₀(Ω), solving
Δw = 0 weakly, ∫_Ω ⟨∇w, ∇v⟩ = 0 for every v ∈ H¹₀(Ω), such that:
wagrees almost everywhere onΩwith a function harmonic onΩ;- if
∫_Ω ⟨∇u, ∇v⟩ = ⟪G, ∇v⟫_{L²}for everyv ∈ H¹₀(Ω), then‖∇(u - w)‖_{L²(Ω)} ≤ ‖G‖_{L²(Ω)}.
The replacement is unique, by
UniformlyEllipticOn.existsUnique_sub_mem_w1p0Submodule_forall_energyFormH1_eq_of_subset_ball.