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Given a domain $\Omega \subset\Bbb R^n$, consider the set $D := \{u \in L^2(\Omega)| \Delta u \in L^2(\Omega)\}$, where $-\Delta$ is the Laplacian. I think this is the domain of the adjoint of $-\Delta$.

My question: is the set $D$ always identical to the Sobolev space $H^2(\Omega)$ ?

Perhaps this is dependent on the boundedness or smoothness of $\Omega$. I didn't found anything about this problem in functional analysis textbooks. Are such sources known to anybody ?

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  • $\begingroup$ you can find such things in Taylor's book Partial Differential Equations I amazon.com/Partial-Differential-Equations-Mathematical-Sciences/… $\endgroup$
    – shu
    Commented Jan 16, 2014 at 13:28
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    $\begingroup$ The book of Renardy-Rogers gives you also information on this. If the boundary is smooth, then yes, in general no. $\endgroup$
    – András Bátkai
    Commented Jan 16, 2014 at 13:30
  • $\begingroup$ When you speak of adjoint of $-\Delta$ you first need to specify the domain of $-\Delta$. How do you define the domain of the Laplacian? $\endgroup$
    – Liviu Nicolaescu
    Commented Jan 16, 2014 at 14:34
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    $\begingroup$ It u and $\Delta u$ are in $L^2$, it certainly does not follow that $u\in H^2$, unless some restriction is imposed at the boundary. $\endgroup$
    – Michael Renardy
    Commented Jan 16, 2014 at 14:47
  • $\begingroup$ Thanks to all for the helpful answers. With respect to the question of Liviu, the domain of the Laplacian should be the space of smooth functions with compact support. $\endgroup$
    – khoefli
    Commented Jan 17, 2014 at 11:17

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