Density of a functional space

Question

Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$: $$\{(f,f\rvert_{\partial D}) : f\in C^\infty(\overline{D}), (\Delta f)\rvert_{\partial D}=0\}.$$ I tried to use density of $C_c^\infty(D)$ in $L^2(D)$, but I didn't get a final answer. This question was motivated by the density of the subspace $$\{(f,f\rvert_{\partial D}) : f\in C^\infty(\overline{D})\}.$$ Thank you for any hint.

Answer 1

I believe that Daniele Tampieri's idea of using completeness Fichera's Theorem is the right one. However I think there is a simpler proof. In view of the Hahn-Banach Theorem, proving the density of the space \begin{equation} \left\{(f,f|_{\partial D})\ |\ f\in S \right\}, \quad \text{where } S=\left\{ f\in C^{\infty}(\overline{D})\ |\ \Delta f|_{\partial D} =0 \right\}, \end{equation} in $L^{2}(D)\times L^{2}(\partial D)$ is equivalent to show that, if $(F,G)\in L^{2}(D)\times L^{2}(\partial D)$ is such that \begin{equation} \int_{D}F\, f\, dx + \int_{\partial D}G\, f\, d\sigma = 0 \end{equation} for any $f\in S$, then $F=0$ a.e. in $D$ and $G=0$ a.e. on $\partial D$. So suppose that these orthogonality conditions hold. In particular we have $$ \int_{D}F\, f\, dx = 0 $$ for any $f\in C^{\infty}_{c}(D)$. It is well known that this implies $F=0$ a.e. in $D$. Therefore $$ \int_{\partial D}G\, f\, d\sigma = 0 $$ for any $f\in S$. Harmonic polynomials obviously belong to $S$ and then $$ \int_{\partial D}G\, \omega\, d\sigma = 0 $$ for any harmonic polynomial $\omega$. Fichera's theorem implies $G=0$ a.e. on $\partial D$.

Answer 2

This is not an answer, but an elaboration of my comments above.

Let $(f,g)\in L^2(D)\times L^2(\partial D)$. Find $g_n\in C^\infty(\partial D)$ such that $g_n\to g$ in $L^2(\partial D)$. Let $h_n\in C^\infty(\bar D)$ be the harmonic extension of $g_n$ to $\bar D$, i.e., the solution of the Dirichlet problem $\Delta h_n=0$ on $D$ and $h_n=g_n$ on $\partial D$. By classical results such an $h_n$ exists and is unique. I will next assume that

(1) Harmonic extension is a bounded map from $L^2(\partial D)$ to $L^2(D)$.

I believe this is true, at least when $\partial D$ is smooth enough, and I think the book The Dirichlet Problem with L2-Boundary Data for Elliptic Linear Equations will be of help, but being far from an expert in elliptic equations with nonsmooth boundary values, I have yet to assemble a proof from a brief glimpse of the book so far. Maybe someone else can help?

Now let's just assume (1), and continue our proof. Using $g_n\to g$ in $L^2(\partial D)$ and (1), we know that $h_n$ converges to some $h\in L^2(D)$. Then $f-h\in L^2(D)$, and we can find $F_n \in C_c^\infty(D)$ such that $F_n\to f-h$ in $L^2(D)$. Now let $f_n=h_n+F_n$. Then $f_n\in C^\infty(\bar D)$, $f_n\to f$ in $L^2(D)$, $f_n|_{\partial D}=h_n|_{\partial D}=g_n\to g$ in $L^2(\partial D)$, and $\Delta f_n|_{\partial D}=\Delta F_n|_{\partial D}=0$.

P.S. It is likely that there is a simpler proof than consulting the Springer book.

Answer 3

The answer is affirmative and as a consequence of the following old result proved by Gaetano Fichera in [4] (see also the survey [2] pp. 54-59).

Theorem (Fichera [4]). Let $D$ be a bounded domain in $\Bbb R^n$, $n\ge 2$, with $C^2$ boundary $\partial D$ and such that $\Bbb R^n\setminus\overline{D}$ is connected. Denoting with $\{\omega_k\}_{k\in\Bbb N}$ the sequence of homogeneous harmonic polynomials, the following three properties hold

1. $\{\omega_k\}_{k\in\Bbb N}$ is complete in $L^2(\partial D)$
2. $\{\partial_\nu\omega_k\}_{k\in\Bbb N}$ is complete in the space $$ \left\{v\in L^2(\partial D)\,\,\Bigg|\, \int\limits_{\partial D}v\mathrm{d}\sigma=0\right\} $$
3. Given a partition $\Sigma_1, \Sigma_2$ of $\partial D$ (i.e. couple of subsets of $\partial D$ such that $\Sigma_1\cap\Sigma_2=\emptyset$ and $\Sigma_1\cup\Sigma_2=\partial D$), the sequence $\big\{(\omega_k,\partial_\nu \omega_k)\big\}_{k\in\Bbb N}$ is complete in $L^2(\Sigma_1)\times L^2(\Sigma_2)$.

Note. The theorem is stated and proved for domains with $C^2$ boundaries, but Fichera's proof is easily adapted to domains with Lyapunov boundaries, i.e. $C^{1,\alpha}$ boundaries, and by using the methods described by Cialdea ([1], [2]), it can be finally extended to domains with $C^1$ boundaries.

Now, defining $$ \begin{align} \mathscr{C_H^\infty}(\overline{D})&=\big\{ f\in C^\infty(\overline{D}): (\Delta f)\rvert_{\partial D}=0\big\}\\ \operatorname{diag}\mathscr{C_H^\infty}(\overline{D})\times&\mathscr{C_H^\infty}(\overline{D})= \big\{(f,f|_{\partial{D}}): f\in\mathscr{C_H^\infty}(\overline{D})\big\} \end{align} $$ as the space of infinitely smooth function with harmonic trace on a domain $D$, by using Fichera's theorem result, for each $(h,g)\in L^2(D)\times L^2(\partial D)$ it is possible to construct a sequence $$ \big\{(f_n,f_n|_{\partial D})\big\}_{n\in\Bbb N}\Subset \operatorname{diag}\mathscr{C_H^\infty}(\overline{D})\times\mathscr{C_H^\infty}(\overline{D}),$$ converging to it.

Step 1. Construct an open cover of the domain $D$ such that the open cover $\{U_n\}_{n\in\Bbb N}$ satisfies the following conditions $$ \begin{cases} D_n\Subset D&\\ \operatorname{dist}(D_n,D) \ge \dfrac{\epsilon}{2^n}&\text{for a properly chosen and fixed }\epsilon >0 \\ \end{cases} $$ where $\operatorname{dist}(A,B)$ is the euclidean distance between the two sets $A, B\in\Bbb R^n$.

Step 2. Define $v_n=\sum_{k=0}^n a_k\omega_k$, where $a_k\in\Bbb R$, $k\in\Bbb N$ are the Fourier coefficients of the expansion of $g$ respect to the complete system of homogeneous harmonic polynomials $\{\omega_k\}_{k\in\Bbb N}$ and, as such, $$ \lim_{n\to\infty}\Vert v_n -g\Vert_{L^2(\partial D)} = 0 $$ Now put $$ f_n(x)=v_n(x)+\int\limits_{D_n}\psi_{\small\frac{\epsilon}{2^{n+1}}}\big(x-y\big)\big[h(y)-v_n(y)\big]\mathrm{d}y\quad\forall n\in\Bbb N $$ where $\psi_\cdot$ is the standard mollifier in $\Bbb R^n$: the sequence $\{f_n\}_{n\in\Bbb N}$ defines the sequence $\big\{(f_n,f_n|_{\partial D})\big\}_{n\in\Bbb N}$ which clearly satisfies the following limit requirement $$ \lim_{n\to\infty}\big\Vert(f_n,f_n|_{\partial D})-(h,g)\big\Vert_{L^2(D)\times L^2(\partial D)}=0\quad \forall (h,g)\in L^2(D)\times L^2(\partial D). $$

Notes

• The intuition of Fang Zheng was the spark that triggered my proof: while difficult to apply as it was stated, his correct suggestion made me remember of the completeness in the sense of Picone for polynomial solutions of PDEs, and the circle of ideas that was developed by his school, notably Gaetano Fichera and his pupil Alberto Cialdea. In particular, the papers [1], [2] and [3] give state of the art results as well as an historical survey and relevant references. Apart from this, this kind of procedure is common when using the variational approch for the solution of PDE problems, since it avoids the Prym-Hadamard phenomenon.
• Edit: Fichera's theorem does not require $D$ nor $\Bbb R^n\setminus\overline{D}$ to be connected: Fichera himself ([4] p. 2) precises this by describing $D$ as a domain with $p$ holes, $p\in\Bbb N$.
• Statement 1 and 2 of Fichera's theorem are easily extendible to the space $L^p(\partial D)$, $1\le p\le+\infty$. For statement 3, things are different: while holding true for any $1\le p\le 2$ as implied by the $p=2$ proof, its validity for $p>2$ is still an open problem. However, this implies that $$ \operatorname{diag}\mathscr{C_H^\infty}(\overline{D})\times\mathscr{C_H^\infty}(\overline{D})\text{ is dense in }L^p(D)\times L^p(\partial D) $$
• Note that $\Delta f_n\neq 0$ for all $(h,g)\in L^2(D)\times L^2(\partial D)$ (or $L^p(D)\times L^p(\partial D)$, $1\le p\le\infty$) for all $x\in\overline{D}$ but only in a neighborhood of $\partial D$. Requiring $f_n$ to be harmonic on the whole $D$ implies the failure of the density property as simple counterexamples can show.

[1] Cialdea, Alberto, Completeness theorems for elliptic equations of higher order with constant coefficients, Georgian Mathematical Journal 14, No. 1, 81-97 (2007). MR2323374, Zbl 1135.42333.

[2] Cialdea, Alberto, Completeness theorems in the uniform norm connected to elliptic equations of higher order with constant coefficients, Analysis and Applications (Singapore) 10, No. 1, 1-20 (2012). MR2876933, Zbl 1243.42043.

[3] Cialdea, Alberto, Completeness Theorems: an example of the legacy of Gaetano Fichera, in: C. Sbordone (Ed.) Equazioni a derivate parziali nell'opera di Gaetano Fichera, Quaderno n. 60 Accademia Pontaniana, Giannini Editore Napoli, 49-68 (2014).

[4] Fichera, Gaetano, Teoremi di completezza sulla frontiera di un dominio per taluni sistemi di funzioni, Annali di Matematica Pura ed Applicata, IV. Serie 27, 1-28 (1948). MR0029014, Zbl 0035.34801.