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I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.

In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary and $p\geq 2$. Let's consider the eigen-system of the Dirichlet Laplacian $\Delta$ on $\mathcal{O}$, there exists numbers $0>\lambda_1>\lambda_2\geq\lambda_3\geq\cdots$ and functions $\{\omega_k\}_{k=1}^\infty$ such that $\{\omega_k\}_{k=1}^\infty$ is an orthogonal basis of $H_0^1(\mathcal{O})$ and an orthonormal basis of $L^2(\mathcal{O})$ and $\omega_k$ is the eigenfunction of $\Delta$ with eigenvalue $\lambda_k$. Then we can define the finite-dimensional projection $P_m$ on $L^2(\mathcal{O})$ by $$P_mv \dot{=}\sum_{k=1}^{m}(v,\omega_k)_{L^2}\omega_k\in L^2.$$ In the statement of Theorem 4.4 in this paper, the authors assume that $\partial \mathcal{O}$ is $C^m$ with $m\geq 2$ integer such that $$m\geq \frac{N(p-2)}{2p}.$$ It is claimed in (4.10) of this paper that under this assumption, we have for $v\in L^p(\mathcal{O})\cap H_0^1(\mathcal{O})$ $$ P_mv\rightarrow v \text{ in $L^p(\mathcal{O})\cap H_0^1(\mathcal{O})$ as $m\rightarrow\infty$}. $$ I wonder whether the convergence in $L^p(\mathcal{O})$ is true?

I think the following may be related to the answer. It has been claimed at the begining of section 5 in https://projecteuclid.org/journals/annals-of-probability/volume-29/issue-4/Stochastic-two-dimensional-euler-equations/10.1214/aop/1015345773.full that there exist a set of Schauder basis $\{e_k\}_{k=1}^\infty\subseteq L^2(\mathcal{O})$, an orthogonal projection $Q_m:L^2(\mathcal{O})\rightarrow span\{e_1,...,e_m\}$ and constant $C>0$ independent of $m$ such that $$\|Q_mv\|_{L^p(\mathcal{O})}\leq C\|v\|_{L^p(\mathcal{O})}.$$

I also wonder whether the family $\{\omega_k\}_{k=1}^\infty$ of eigenfunctions of Dirichlet Laplacian is a Schauder basis in ${L^p(\mathcal{O})}$.

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    $\begingroup$ Something in the paper is not clear to me, too. It is true that $P_m v \to v$ in $H^1$ and then in $L^p$ if $2\leq p \leq 2*$ (assuming $v \in H_0^1$). The same holds in $H^{2m}$ (plus some b.c.) using elliptic regularity and here the regularity of the boundary enters) but assuming at least $v \in H^{2m}$ and then in $L^p$ for those $p$ for which the Sobolev embedding holds. But I doubt more. Concerning the general question, this is not true. Spherical harmonics in several dimensions are a basis in $L^p $ only for $p=2$. $\endgroup$
    – Giorgio Metafune
    Commented Sep 22, 2023 at 11:39
  • $\begingroup$ @GiorgioMetafune Thank you for your answer, it helps me a lot. I think the question I asked is not true in general. The eigenfunctions of Dirichlet Laplacian cannot be a Schauder basis in $L^p(\mathcal{O})$. Once it is a Schauder basis, by approximating by functions in $H^{2m}$, the limit can be shown... I'm also interested in the boundary condition you just mentioned. Is there any reference showing how the boundary condition is used in showing the limit? $\endgroup$
    – ze min jiang
    Commented Sep 22, 2023 at 13:38
  • $\begingroup$ @GiorgioMetafune May I ask one more question? at the begining of section 5 in projecteuclid.org/journals/annals-of-probability/volume-29/…, the authors said that they can take a family $\{e_n\}_{n=1}^\infty\subseteq W^{2,2}(\mathcal{O})\cap W^{2,p}(\mathcal{O})$ which is an orthonormal basis in $H^1_0(\mathcal{O})$ and a Schauder basis in $W^{1,p}(\mathcal{O})$, how can I achieve this? (Here, the domain $\mathcal{O}$ is unbounded with smooth boundary) $\endgroup$
    – ze min jiang
    Commented Sep 22, 2023 at 13:39
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    $\begingroup$ @GiorgioMetafune I think what they need is unrelated to the eigenfunctions. A smooth version of Haar system sounds like a good angle. Thanks a lot anyway. $\endgroup$
    – ze min jiang
    Commented Sep 22, 2023 at 14:15
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    $\begingroup$ @GiorgioMetafune I think the first question is finally addressed. I believe that the authors has lost the condition $v_\tau\in H^m$. Thanks a lot for your interpretation in how the b.c is introduced. $\endgroup$
    – ze min jiang
    Commented Sep 22, 2023 at 16:44

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