Smooth functions are dense in W^{k,p}(Ω) (Meyers–Serrin) #
For 1 ≤ p < ∞, every natural order k, and an arbitrary open set Ω of a
finite-dimensional real inner product space, the elements of W^{k,p}(Ω) with a representative
which is smooth on Ω are dense in W^{k,p}(Ω), in the full Sobolev norm. In other words
H = W: the closure of C^∞(Ω) ∩ W^{k,p}(Ω) is all of W^{k,p}(Ω). No boundedness and no
regularity of the boundary of Ω is assumed.
The approximants are smooth on Ω but need not be smooth up to its boundary, nor compactly
supported in Ω: on standard nontrivial bounded domains (for example, Euclidean balls in positive
dimension), W^{k,p}_0(Ω) is a proper subspace for k ≥ 1, so test functions are not dense. On
the whole space, density of globally smooth representatives is
TauCeti.Wkp.dense_contDiff_representatives.
Cutoffs #
The proof localizes u by a decomposition of unity, which needs the product ζ u of u with a
smooth function ζ compactly supported in Ω. This file shows that such a product is a
W^{k,p}_0(Ω) function (TauCeti.Wkp.exists_mem_wkp0Submodule_value_ae_eq_mul). Take an open
U with compact closure in Ω containing the support of ζ, and test functions ψ n on Ω
whose restrictions converge to u in W^{k,p}(U)
(TauCeti.Wkp.exists_testFunction_approximation_restrictL). The Leibniz estimate
TauCeti.Wkp.norm_ofTestFunctionₗ_le_of_eqOn_mul on U bounds the W^{k,p} norm of the test
function ζ (ψ n - ψ m) by that of ψ n - ψ m on U, and extension by zero from U to Ω
preserves it. So the test functions ζ ψ n converge in W^{k,p}(Ω), and their limit has value
ζ u.
The gluing #
Take a decomposition of unity (ζ j) of Ω by test functions, together with cutoffs χ j equal
to one on the support of ζ j and locally finite in Ω
(IsOpen.exists_contDiff_decomposition_cutoff). Each piece ζ j u is approximated, to within
ε 2^{-j}, by a test function ζ j ψ j, which vanishes wherever χ j does. The sum
f = ∑ j, ζ j ψ j is locally finite in Ω, hence smooth there.
The series ∑ j, ζ j u need not converge to u in W^{k,p}(Ω), so the comparison with u is
made through the corrections instead. These are absolutely summable in the Banach space
W^{k,p}(Ω), to some w of norm at most ε / 2; after passing to a subsequence,
representatives of their partial sums converge almost everywhere on Ω to f - u. Hence
u + w is within ε of u, and its value is represented by f.
Main declarations #
TauCeti.Wkp.exists_mem_wkp0Submodule_value_ae_eq_mul: the product of aW^{k,p}(Ω)function with a smooth function compactly supported inΩlies inW^{k,p}_0(Ω).TauCeti.Wkp.exists_contDiffOn_approximation: everyu ∈ W^{k,p}(Ω)is a Sobolev-norm limit of elements with representatives smooth onΩ.TauCeti.Wkp.dense_contDiffOn_representatives: those elements are dense inW^{k,p}(Ω).
References #
- N. G. Meyers, J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 1055–1056.
- L. C. Evans, Partial Differential Equations, §5.3.2, Theorem 2.
Cutoffs of Sobolev functions have zero boundary values. If 1 ≤ p < ∞, u ∈ W^{k,p}(Ω)
and ζ is smooth with compact support inside Ω, then the product ζ u lies in
W^{k,p}_0(Ω): some element of W^{k,p}_0(Ω) has value ζ u almost everywhere on Ω. No
regularity of Ω is assumed.
Meyers–Serrin approximation in W^{k,p}(Ω). For 1 ≤ p < ∞, every order k and an
arbitrary open set Ω, every u ∈ W^{k,p}(Ω) is a limit, in the full Sobolev norm, of elements
whose values are represented by functions smooth on Ω. No boundedness or boundary regularity of
Ω is assumed.
Meyers–Serrin: H = W in W^{k,p}(Ω). For 1 ≤ p < ∞, every order k and an
arbitrary open set Ω, the elements of W^{k,p}(Ω) represented by functions smooth on Ω are
dense in W^{k,p}(Ω), in the full Sobolev norm.