Re: Interpolation between (modified) Sobolev Spaces

This is well studied, including the case a=1/2, which leads to a
special kind
of space. The books of Lions and Magenes are a good reference.


On Aug 23, 12:30 pm, Ilya Zakharevich <nospam-ab...@xxxxxxxxx> wrote:
Since there is some ambiguity in naming convention: here Sobolev spaces
mean 1-index, Hilbert, Sobolev spaces (denoted H^s, or W^2_s).

Spaces H^s form a linear interpolation scale: interpolation between
H^s and H^t is H^r with r = a t + (1-a) s. However: what happens if
one modifies one of these spaces by a finite-dimensional space?

For example: Take L_2 = H^0 as one end of interpolation, and take the
vector subspace of H^1 of codimension 1, consisting of vanishing at 0
functions as another end. Interpolate between these spaces. What happens?

The obvious conjecture is that for a < 1/2 one gets H^a, for a > 1/2
one gets a similar hyperplane {f(0)=0} inside H^a. If one is very
bold conjecturer, one could extend this conjecture to a=1/2 too: one
gets H^{1/2}.

Frankly speaking, I'm pretty sure that the behaviour with a not 1/2
holds; but I have no idea what actually happens at 1/2. Is it
investigated already? Is there some simple way to calculate I'm
overlooking?

Thanks,
Ilya



.

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