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.

  • $\begingroup$ There is a answer to a related question with some info, maybe this is sufficiently elegant.. $\endgroup$
    – Hannes
    Jul 29 '20 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 '20 at 19:28

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.

  • $\begingroup$ 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 '20 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 at 2:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.