Master interior difference-quotient energy estimate #
The interior second-derivative estimate (Evans, Partial Differential Equations (2nd ed.),
§6.3.1; Gilbarg-Trudinger, Elliptic PDE of Second Order, Theorem 8.8) proceeds by testing
the weak formulation of L u = f with the difference-quotient test element
v_h = -Dₖ^{-h}(ξ² Dₖ^h u), using discrete integration by parts to move the outer difference
quotient onto the coefficient factor, uniform ellipticity from below to control the leading
term, and Cauchy-Schwarz together with the Peter-Paul (Young) inequality to absorb the
commutator, cross, zeroth-order, and right-hand terms.
Main declarations #
evansTest: the admissible test elementv_h = -Dₖ^{-h}(ξ² Dₖ^h u) ∈ H₀¹(Ω), whose membership is two applications ofcutoffMul_diffQuotG_mem_H01.interior_diffQuot_energy_bound: the master energy estimate the sections below assemble.
Admissible Evans test element #
Admissible Evans test element v_h = -Dₖ^{-h}(ξ² Dₖ^h u) ∈ H₀¹(Ω). The inner
cutoff ξ² and the outer cutoff θ (which is ≡ 1 on tsupport ξ) localise the two
difference quotients so that the composite stays inside H₀¹(Ω); membership is two
applications of the crux admissibility lemma cutoffMul_diffQuotG_mem_H01, together with
closure of the submodule under negation. This is the single admissible test vector that the
weak formulation consumes in the difference-quotient energy method (Evans, Partial
Differential Equations (2nd ed.), §6.3.1).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The ambient-graph value of evansTest is the negated cutoff of the outer difference
quotient of ξ² Dₖ^h u.
Support control and θ-invisibility #
Discrete integration by parts #
Support of the inner cutoff data and the Evans coordinate reduction #
Extension-by-zero weak derivative and the first-order global energy #
Extension by zero of an H₀¹ element preserves the weak gradient. For u ∈ H₀¹(Ω), the
whole-space extension by zero of the function value u₀ has whole-space L² weak
k-derivative equal to the extension by zero of the gradient component u_{k+1}. Because u
vanishes at the boundary (it lies in the closure of the compactly supported test functions), no
boundary term appears when integrating against an arbitrary whole-space test function φ: the
identity ∫ (extendL2 u₀) ∂ₖφ = -∫ (extendL2 u_{k+1}) φ is closed under L² limits and holds
on every test-function graph by classical integration by parts, hence on all of H₀¹(Ω) (Evans,
Partial Differential Equations (2nd ed.), §5.8.2).
First-order global energy estimate. For a weak solution u ∈ H₀¹(Ω) of
L u = f whose transport field b vanishes and whose zeroth-order coefficient c is
nonnegative (a.e. on Ω), the full gradient energy is bounded by the data:
λ ∑ᵢ ‖u_{i+1}‖² ≤ ‖f‖ · ‖u₀‖. Testing the weak formulation with u itself, ellipticity
bounds the principal part from below, the transport term drops (b = 0) and the
zeroth-order term has a sign (c ≥ 0), so only the right-hand pairing ⟪f, u₀⟫ survives
(Evans, Partial Differential Equations (2nd ed.), §6.2.2).
Ambient-space generalisation of norm_diffQuotD_u0_le (proof of concept). H01 is not
needed for this inequality itself: it is needed only to manufacture the one whole-space
weak-derivative fact hasWeakDeriv_extendL2_of_mem_H01 supplies. Factoring that fact out as a
hypothesis hW shows the rest of norm_diffQuotD_u0_le's proof (the identity diffQuotD = restrict ∘ diffQuot ∘ extendL2, the non-expansive restriction, and the
difference-quotient/weak-derivative bound norm_diffQuot_le_of_hasWeakDeriv) survives unchanged
for an arbitrary ambient graph U : H1amb Ω, with no reference to H01 at all. Evans, Partial
Differential Equations (2nd ed.), §6.3.1, Remark (i), states that the zero-trace hypothesis
this lemma's H01-typed sibling states is not required for the interior estimate; this
declaration isolates that the only place it was doing work in this particular step was in
supplying hW, not in the inequality.
Kept as that record. The chain runs on the H01-typed sibling, so nothing consumes this
one.
Master assembly toolkit #
Master interior difference-quotient energy estimate #
First-order gradient bound. Each gradient component of a weak solution is bounded in
L² by the data: ‖∂ᵢu‖ ≤ √((1 + 4γ) / (2λ)) (‖f‖ + ‖u₀‖), where γ is the Gårding shift
constant, through which the transport and zeroth-order coefficients enter. This is the
first-order energy estimate firstOrder_energy_le combined with the arithmetic-geometric
mean inequality.
Master interior difference-quotient energy estimate. For a W^{1,∞}-coefficient
weak solution u ∈ H₀¹(Ω) of L u = f, an inner cutoff ξ and an
outer cutoff θ ≡ 1 on the shift-reachable part of tsupport ξ², the cutoff-weighted energy
of the interior difference quotient of the gradient is bounded by the data, uniformly in the
step h: (λ/2) ∑ᵢ ‖ξ · Dₖ^h ∂ᵢu‖² ≤ C (‖f‖² + ‖u₀‖²). The constant is quantified before
the solution and the datum, so it depends only on
λ, Λ, A₁, d, γ, ‖b‖∞, ‖c‖∞, ‖ξ‖∞, ‖∂ξ‖∞, and on none of u, f, h. Testing the weak
formulation with the
admissible Evans element v_h = -Dₖ^{-h}(ξ² Dₖ^h u), discrete integration by parts
(evansTest_bilin_L2D) moves the outer difference quotient onto the coefficient action; the
discrete Leibniz split (norm_diffQuotD_actL_sub_le) exposes the translated-coefficient
leading term, controlled from below by ellipticity (energy_ge for the translate, same λ);
Cauchy-Schwarz and the Peter-Paul inequality absorb the commutator, cross, transport,
zeroth-order and right-hand terms, five families each spending an eighth of the ellipticity
lower bound, with the first-order energy bound firstOrder_energy_le supplying all gradient
data
(Evans, Partial Differential Equations (2nd ed.), §6.3.1; Gilbarg-Trudinger, Elliptic
PDE of Second Order, Theorem 8.8).