By Luc Tartar
After publishing an advent to the Navier–Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with one other set of lecture notes in line with a graduate path in components, as indicated through the name. A draft has been to be had on the net for many years. the writer has now revised and polished it right into a textual content obtainable to a bigger audience.
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Extra info for An Introduction to Sobolev Spaces and Interpolation Spaces
For showing this, one must observe that if T is a distribution of order 0, then one can extend the deﬁnition of T, ϕ to test functions ϕ ∈ Cc (Ω) instead of ϕ ∈ Cc∞ (Ω), by choosing a special smoothing sequence ε and then deﬁning T, ϕ = limε→0 T, ϕ ε , with ε > 0 small enough in order to have support(ϕ) + support( ε ) ⊂ Ω. 4 Truncation; Radon Measures; Distributions 19 every compact K ⊂ Ω one has K |fn (x)| dx → 0. One identiﬁes any function f ∈ L1loc (Ω) with a Radon measure (and therefore with a distribution), which one usually also writes f , deﬁned by the formula f (x)ϕ(x) dx for all ϕ ∈ Cc (Ω).
If T ∈ D (Ω) and ψ ∈ C ∞ (Ω), then ψ T (or T ψ) is the distribution deﬁned by ψ T, ϕ = T, ψ ϕ for all ϕ ∈ Cc∞ (Ω). 10) Notice that one only deﬁnes the product of a distribution by a function in C ∞ (Ω) (or more generally of a distribution of order ≤ m by a function in C m (Ω)), but ﬁrst one must check that ψ T is a distribution, and this follows from Leibniz’s formula. 11) n n! (n−m)! ; it is easily proven by induction, starting from (f g) = f g + f g . Writing a generalization of Leibniz’s formula to the N -dimensional case is simpliﬁed by using a notation for multi-indices.
11) n n! (n−m)! ; it is easily proven by induction, starting from (f g) = f g + f g . Writing a generalization of Leibniz’s formula to the N -dimensional case is simpliﬁed by using a notation for multi-indices. 4. If α = (α1 , . . , αN ), then |α| means |α1 | + . . + |αn |, β ≤ α means βj ≤ αj for j = 1, . . , N , α! means α1 ! . , and α β means α1 αN α! . = . (α−β)! 12) and it is easily proven by induction on N . β α−β ∞ ϕ||∞ if support(ϕ) One has ||Dα (ψ ϕ)||∞ ≤ β≤α α β ||D ψ||L (K) ||D is contained in a compact K ⊂ Ω, so that one has max|α|≤m ||Dα (ψ ϕ)||∞ ≤ C(K) max|α|≤m ||Dα ϕ||∞ ; one also deduces that if T is a distribution of order ≤ m, then ψ T is also a distribution of order ≤ m.