By Luc Tartar
After publishing an creation to the Navier–Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with one other set of lecture notes in response to a graduate path in elements, as indicated by way of the identify. A draft has been on hand 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
4) Ω It is not really such a good notation, because it relies upon having selected the Lebesgue measure dx and it would be better to call this measure (or distribution) f dx; this abuse of notation is of no consequence for open sets of RN , and corresponds to the usual identiﬁcation of L2 (Ω) with its dual, but when one deals with a diﬀerentiable manifold one should remember that there is no preferred volume form like dx. 5) and it is a Radon measure (and therefore a distribution). If a sequence an ∈ Ω converges to the boundary ∂Ω of Ω and cn is an arbitrary sequence, then µ = n cn δan is a Radon measure in Ω because in the formula µ, ϕ = c ϕ(a n n ), only a ﬁnite number of an belong to the compact support K n of ϕ.
6. W0m,p (Ω) is the closure of Cc∞ (Ω) in W m,p (Ω). 6 Density of Tensor Products; Consequences 31 If the boundary ∂Ω is Lipschitz, then the functions of W0m,p (Ω) are 0 on the boundary, as will be seen later. If ∂Ω is too small, then it may happen that W0m,p (Ω) = W m,p (Ω); this is related to the fact that the functions in W m,p (Ω) are not necessarily continuous, and Sobolev’s embedding theorem, given now and proven later, tells for what values of m and p functions in W m,p (Ω) are automatically continuous.
In order to show that K |f n − f |q dx → 0 for a compact K and 1 ≤ q < ∞, one notices that the integral only uses values of f in a ball centered at 0 with radius R0 large enough (for the ball to contain K and K + support( ˇn )), so that if f coincides with f inside the ball centered at 0 with radius R0 and is 0 outside it, then the integral is K |f n − f |q dx, which does converge to 0 as n → ∞ because f n converges to f in Lq (RN ) as n → ∞, as a consequence of the ﬁrst part and of the fact that f belongs to Lq (RN ).
An Introduction to Sobolev Spaces and Interpolation Spaces by Luc Tartar