Interior H² regularity for a constant principal coefficient #
Let A be a constant, uniformly elliptic coefficient matrix and let u ∈ H¹(Ω) be a weak
solution of the divergence-form equation
-div(A ∇u) + ⟨b, ∇u⟩ + c u = f in Ω, with f ∈ L²(Ω) and b, c ∈ L∞(Ω),
meaning ∫_Ω (⟨∇v, A ∇u⟩ + ⟨b, ∇u⟩ v + c u v) = ∫_Ω f v for every v ∈ H¹₀(Ω).
No boundary condition is imposed on u and nothing is assumed about ∂Ω. This file proves that
u ∈ H²_loc(Ω): on every open V whose closure is compact and contained in Ω, the restriction
of u is the first-order part of an element of W^{2,2}(V).
Localization #
The lower-order terms belong to L², so moving them to the right-hand side gives
-div(A ∇u) = g with g = f - ⟨b, ∇u⟩ - c u ∈ L²(Ω). No sign or smallness condition on b
or c is needed. The whole-space theorem
TauCeti.PDE.UniformlyEllipticOn.exists_lowerOrder_eq does the analytic work; what is proved here
is that a cutoff of a local solution is a global one. For ψ smooth and compactly supported in
Ω, the product ψ u, extended by zero, lies in H¹(ℝⁿ) and solves
-div(A ∇(ψ u)) = ψ g - ⟨∇u, A ∇ψ⟩ - ⟨∇ψ, A ∇u⟩ - u div(A ∇ψ) on ℝⁿ,
whose right-hand side is again in L², because every term carrying u or ∇u also carries a
bounded, compactly supported factor built from ψ. The identity is checked against test
functions φ on ℝⁿ, which suffices by density
(TauCeti.PDE.isWeakSolutionDirichlet_iff_forall_testFunction). The term ψ ∇u is handled by
testing the equation for u against ψ φ ∈ H¹₀(Ω), and the term u ∇ψ by the definition of the
weak derivative of u, tested against the components of φ A ∇ψ, which are test functions on
Ω. Choosing ψ = 1 near closure V makes ψ u agree with u on V.
Main declarations #
TauCeti.PDE.divMatrixGradientandTauCeti.PDE.localizedForcing: the divergence of the conormal cutoff field and the forcing term in the localized equation.TauCeti.PDE.exists_isWeakSolutionDirichlet_extendByZeroL_contDiffSMul: a cutoff of a weak solution, extended by zero, is a weak solution on the whole space with an explicitL²forcing term.TauCeti.PDE.exists_isWeakSolutionDirichlet_top_ae_eq_on_of_isCompact: near a compact subset ofΩ, a weak solution and its forcing agree with a whole-space weak solution and its forcing.TauCeti.PDE.UniformlyEllipticOn.exists_lowerOrder_eq_restrictL: interiorH²regularity.
References #
- L. C. Evans, Partial Differential Equations, §6.3.1, Theorem 1 (interior
H²regularity). - D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 8.8.
The divergence div(A ∇ψ) = ∑ᵢ ∂ᵢ (A ∇ψ)ᵢ of the conormal field of ψ, computed in the
standard basis. It is the zeroth-order coefficient that commuting the operator -div(A ∇ ·) past
a cutoff ψ produces.
Equations
- One or more equations did not get rendered due to their size.
Instances For
div(A ∇ψ) vanishes off the support of ψ.
div(A ∇ψ) is continuous for smooth ψ.
The right-hand side of the equation satisfied by a cutoff ψ u of a weak solution u of
-div(A ∇u) = f:
-div(A ∇(ψ u)) = ψ f - ⟨∇u, A ∇ψ⟩ - ⟨∇ψ, A ∇u⟩ - u div(A ∇ψ).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The localized forcing term is square integrable, since each of its terms is an L²(Ω)
function times a bounded one.
A cutoff of a weak solution solves an equation on the whole space. Let u ∈ H¹(Ω) be a
weak solution of -div(A ∇u) = f in Ω for a constant matrix A, with f ∈ L²(Ω) and no
boundary condition, and let ψ be smooth and compactly supported in Ω. Then ψ u, extended by
zero, is a weak solution of -div(A ∇w) = g on the whole space for g ∈ L²(ℝⁿ) given almost
everywhere by
g = ψ f - ⟨∇u, A ∇ψ⟩ - ⟨∇ψ, A ∇u⟩ - u div(A ∇ψ), extended by zero.
No ellipticity and no regularity of ∂Ω is needed.
Localizing a weak solution to the whole space. Let u ∈ H¹(Ω) be a weak solution of
-div(A ∇u) = f in Ω for a constant matrix A, with f ∈ L²(Ω) and no boundary condition.
Near any compact S ⊆ Ω, u agrees, in value and in gradient, with a weak solution
w ∈ H¹(ℝⁿ) of an equation -div(A ∇w) = g on the whole space, with g ∈ L²(ℝⁿ)
and g = f almost everywhere on S.
One may take for w the product of u with a smooth cutoff equal to one near S and compactly
supported in Ω, extended by zero
(TauCeti.PDE.exists_isWeakSolutionDirichlet_extendByZeroL_contDiffSMul). This is the device
that reduces interior regularity to regularity on the whole space.
Interior H² regularity for a constant principal coefficient. Let A be a constant,
uniformly elliptic matrix, let b, c ∈ L∞(Ω), and let u ∈ H¹(Ω) be a weak solution of
-∂ⱼ(Aⁱʲ ∂ᵢu) + ⟨b, ∇u⟩ + c u = f in Ω, with f ∈ L²(Ω),
in the sense that ∫_Ω (⟨∇v, A ∇u⟩ + ⟨b, ∇u⟩ v + c u v) = ∫_Ω f v for every
v ∈ H¹₀(Ω), with no boundary condition on u. Then u ∈ H²_loc(Ω): on every open V
whose closure is compact and contained in Ω, the restriction of u is the first-order part
of an element of W^{2,2}(V).
No regularity of ∂Ω, sign of c, or smallness of the lower-order terms is assumed.