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How one can identify the following (complex) interpolation space $$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$ where $\Omega$ is a regular domaine. After research, it seems that this depend on the position of $\theta \in (0,1)$:

for $0<\theta<1/4$, $E_\theta=H^{2\theta}(\Omega)$ and for $1/4 <\theta <1$ we have $E_\theta=H^{2\theta}_0(\Omega)$, while the case $\theta =1/4$ is critical.

Some inclusions are immediate while the others are not. Is there any elegant way to establish such identification?

Any reference would be helpful.

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  • $\begingroup$ There is a answer to a related question with some info, maybe this is sufficiently elegant.. $\endgroup$
    – Hannes
    Jul 29, 2020 at 7:02
  • $\begingroup$ @Hannes thank you, but this gives just the link to domains of fractional powers and not the explicit characterization depending on $\theta$. $\endgroup$
    – Migalobe
    Aug 17, 2020 at 19:28

1 Answer 1

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This is a result by R. Seeley: Interpolation in Lp with boundary conditions, Studia Mathematica, 1972.

The main ingredient of the proof is that step functions are pointwise multipliers in Hs for s<1/2.

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  • $\begingroup$ Thank you, but it seems that the result was already proved by many others, e.g., Fujiwara, Grisvard... I'm looking for an elementary way to do it. $\endgroup$
    – Migalobe
    Mar 3, 2020 at 6:33
  • $\begingroup$ The book of Lions and Magenes is another reference for this kind of result. But what do you mean by "elementary"? Surely not using pre-calculus only? $\endgroup$ Mar 26, 2021 at 2:48
  • $\begingroup$ @MichaelRenardy I meant not using heavy theories such as $H^\infty$ calculus, or a way that can be generalized to more complex spaces e.g., product spaces. $\endgroup$
    – Migalobe
    Aug 5, 2022 at 1:54

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