Notes on the history of trace theorems on a Lipschitz domain
14 minute read
Background
When writing papers on the theoretical analysis of virtual element methods, my mentor asked me to do some sort of a detective work on how a fractional order norm is defined on the boundary of a domain. So this is a summary, more of a compilation than actual analyses. A trace theorem is in general the following type of theorem:
For a domain $K$ satisfying certain property, a function $g$ in certain space $\mathcal{X}$ defined on $\partial K$ can be continuously extended into $K$, and the extension $Eg\in \mathcal{Y}$ satisfies
$$\|Eg\|_{\mathcal{Y}}\leq c\|g\|_{\mathcal{X}}.$$
In the context of using finite element spaces to approximate the weak formulation of PDEs, the main question is:
For a Lipschitz domain $K$, how should the $H^{1/2}(\partial K)$-seminorm of the trace of an $H^1(K)$ function defined? and how is the constant in the trace theorem (norm equivalence) depending on the shape of $K$?
There are various ways to defined an $H^{1/2}(\partial K)$-norm, for either a continuous $H^1$ function or functions in the discrete approximation spaces living on a mesh:
Quotient norm.
Aronszajn-Gagliardo-Slobodeckij-Sobolev norm as a singular double integral.
Real interpolation norm first appeared in the book of Lions and Magenes.
When there are a mesh, we can have
Scaled Laplace-Beltrami norm by treating the grids on the boundary skeleton as a graph.
Weighted $L^2$-norm when the function satisfying conditions such as being zero mean.
Commonly used conditions
Any $W^{k,p}$ Sobolev function on a domain $K$ can be continuous extended to the
whole space, such domain $K$ has $(s,p)$-extension property.
For each $z\in \partial K$, there exists a cone $C$ such that
$C\backslash \{z\}\subset K$.
The size of the cone does not depend on the location of $z$.
$K$ is locally a Lipschitz hyper-graph.
The cone condition can be found in e.g., Definition 1.2.2.1 (Grisvard, 1985). Notice Lipschitz domain satisfies both cone and extention properties.
Trace theorem or its variants
The most common technique in proving a trace theorem for a Sobolev function on a Lipschitz domain is: first performing a partition of unity, then using the Lipschitz condition to flatten the boundary locally; the problem is tamed to an extension (with explicit construction available) problem on the half plane. Some examples for $H^{1/2}(\partial K) = W^{1/2, 2}(\partial K)$:
Theorem 1.5.1.3 and Theorem 1.5.2.1 in Grisvard’s book (Grisvard, 1985).
Chapter 2 Lemma 5.2 to Theorem 5.7 in Nečas’s book (Nečas, 1967) (translated version
(Nečas, 2011)). The author calls the existence of the extension ``the
converse theorem’’.
Theorem 3.1 and 4.1 in Jonsson-Wallin 1978 paper (Jonsson and Wallin, 1978) with $\alpha = 1$, $\beta = 1/2$, $k=0$.
Existence of a smooth partition of unity for Lipschitz domain in McLean’s book (McLean, 2000) Theorem 3.21.
Theorem 7.5 in Lions-Magenes (Lions and Magenes, 2012).
In a more general setting in the Besov spaces in (Triebel, 2010), Theorem 3.3.3
(using term retraction and co-retraction).
Marshall proved a smoother version in \cite{Marschall1987trace} characterizing $c$ explicitly; Nikol’ski\u{i} et all proved a general version in \cite{Nikol1988weighted} as well.
The techniques of directly bridging the Sobolev norm with Slobodeckij-type norms can
be found in (Maz’ya, 1984)(Yakovlev, 1967).
The most relevant one to my own research is Lemma 2.1 in (Arnold, Scott and Vogelius, 1988), which uses a Hardy inequality to quantify the effect of the interior angle on the constant in the trace inequality.
The following theorem is a special case of Theorem 8.3 in (Lions and Magenes, 2012), and with the extension of (Grisvard, 1985) to bounded Lipschitz domains in Theorem
1.5.1.2.
For $K$ a Lipschitz polygon/polyhedron ($C^{0,1}$-boundary), the mapping $\gamma$
$$
\gamma: H^1(K) \to H^{1/2}(\partial K), \quad u \mapsto \gamma(u)
$$
is surjective, and there exists a continuous linear right inverse $\mathcal{R}$:
$$
\mathcal{R}: H^{1/2}(\partial K) \to H^1(K),\quad g\mapsto \mathcal{R} g,
$$
such that $\gamma \big(\mathcal{R} g\big) = g$. Continuity of the mapping reads:
$$
\|{\mathcal{R} g}\|_{1,K}\leq \|{g}\|_{1/2,\partial K}.
$$
Conversely, for a $g$ such that Slobodeckij norm is finite, it can be constructed in
$K$ a function $u = \mathcal{E}g$ such that:
$$
\|{u}\|_{1,K}\leq \|{g}\|_{1/2,\partial K}.
$$
Outlines of the proofs
For $K$ a Lipschitz polygon/polyhedron, $u\in H^{1/2}(\partial K)$, and
$\int_{\partial K} u \,ds = 0$, then there exists an
extension $Eu\in H^1(K)$ such that
$$
|Eu|_{1,K} \lesssim |u|_{1/2,\partial K}.
$$
An informal outline of the proof is as follows.
For $\{C_k\}_{1\leq k\leq M_K}$ being a finite open cover
of $\partial K$ and each of $C_k$ is a cube in $\mathbb{R}^d$. There exists a partition of unity
((Lions and Magenes, 2012) (7.12), (McLean, 2000) Theorem 3.21 for Lipschitz):
$\{\theta_k\}_{1\leq k\leq M_K}$, such that $\theta_k \in \mathcal{D}(\mathbb{R}^d\cap\partial K)$,
$\sum \theta_k = 1$, and $\mathrm{supp}\, \theta_k \subset C_k$. Consider each
$\theta_k u|_{\partial K}$, $\mathrm{supp}\, (\theta_k u)\subset C_k$.
There exists a map $\phi^*_k: C_k \to \mathcal{C}$ that maps a function supported in
$C_k\cap \partial K$ to a
function supported in $\mathcal{C}\cap \{y_d = 0 \}$ ($d=2,3$), where $\mathcal{C}$ is the following
open cylinder $\mathcal{C}:= \{y = (y',y_d): |y'|<1, -1<y_d<1 \}$, i.e.,
$\mathrm{supp}\, \phi^*_k(\theta_k u) \subset \mathcal{C}\cap \{y_d = 0 \}$. By (Jonsson and Wallin, 1978) Theorem 4.1, there exists an extension $E_k$ such that
$$
\|{E_k(\phi^*_k(\theta_k u))} \|_{1,\mathbb{R}^d} \lesssim
\|{\phi^*_k(\theta_k u) }\|_{1/2,\mathcal{C}\cap \{y_d = 0 \} }.
$$
By the construction using the cut-off function (in the proof of Theorem
4.1 in (Jonsson and Wallin, 1978)), the $\mathcal{C}$ can be always enlarged that the support
of the extension is within $\mathcal{C}$: i.e. we can safely put
$$
\|{E_k(\phi^*_k(\theta_k u))} \|_{1,\mathcal{C}} \simeq
\|{E_k(\phi^*_k(\theta_k u))}\|_{1,\mathbb{R}^d}.
$$
Now the extension on $K$ is just:
$$
E u := \sum_{k=1}^{M_K} (\phi^*_k)^{-1} \big( E_k(\phi^*_k(\theta_ku)) \big).
$$
Now
$$
\begin{aligned}
\|{E u }\|_{1,K} &\leq \sum_{k=1}^{M_K} \|{(\phi^*_k)^{-1} \big(
E_k(\phi^*_k(\theta_ku)) \big)} \|_{1,C_k}
\\
&\lesssim \sum_{k=1}^{M_K} \|{E_k(\phi^*_k(\theta_ku)) }\|_{1,\mathcal{C}}
\\
&\lesssim \sum_{k=1}^{M_K}
\|{\phi^*_k(\theta_k u)}\|_{1/2,\mathcal{C}\cap \{y_d = 0 \} }
\\
&\lesssim \sum_{k=1}^{M_K}
\|{\theta_k u}\|_{1/2,C_k\cap \partial K }
\\
& \lesssim \|{u}\|_{1/2,\partial K}.
\end{aligned}
$$
The last inequality uses both an algebraic Cauchy-Schwarz inequality,
and the non-localness of Slobodeckij-type $1/2$-norm (sum of parts is less than whole). As a
result the constant depends on $M_K^{1/2}$ (number of partitions). The
first inequality from $C_k$ to $\mathcal{C}$ depends on the maximum of the
Jacobian of
$(\phi^*_k)^{-1}$. By the proof of Lemma 1.5.1.9 in (Grisvard, 1985), this
constant depends on $(1+L_{\partial K}^2)^{-1/2}$ ($L^2$ part in this), where $L_{\partial K}$
is the Lipschitz constant of the boundary as a graph of a Lipschitz function. The
second inequality about the extension on the straightened box in $\mathbb{R}^d$ is constant
free. Lastly, using the compactness induced by the extra constraint
$\int_{\partial K} u\,ds = 0$, both norms can be replaced by semi-norms. Thus the Lemma
follows.
Remark on the constant
The size of the covering has to be quasi-uniform. This means we cannot cover each
edge $e_k$ of a polygon with a box $C_{k}$. When an edge’s length goes to zero, the
Jacobian of $(\phi_k)^{-1}$ is gonna blow up. So the Lipschitz constant does go into
the bound.
Remark on the proof in Grisvard’s book
The semi-norm equivalence is used in the lemma above. In order that semi-norm is
used as a norm, a trace inequality can be used.
Based on the proof of the trace inequality in (Grisvard, 1985) Theorem 1.5.1.10,
the constant depends on the $C^1$-norm of the lift mapping, which is troublesome. Here an
improved version is presented using Grisvard’s technique.
For $K$ a bounded Lipschitz polytope in $\mathbb{R}^d$, for any $u\in H^1(K)$,
such that $\int_{\partial K} u\,ds = 0$,
$$
\|u\|_{0,\partial K} \lesssim \|u\|_{0,K}^{1/2} |u|_{1,K}^{1/2}. \tag{T}
$$
An informal proof alongside with comments on the constant is presented as follows.
Without loss of generality we consider a polygon $K \subset \mathbb{R}^2$,
the unit normal vector $\boldsymbol{n}$ to whose boundary can be extended to the interior, say
$\boldsymbol{q}$. The extension $\boldsymbol{q}$ can be obtained by solving
$$
\begin{cases}
-\Delta \phi = f_K & \text{in }K,
\\[1pt]
\partial \phi/\partial n= 1 & \text{on }\partial K.
\end{cases}\tag{N}
$$
Then let $\boldsymbol{q} = \nabla \phi$. For any $u\in H^1(K)$, we have:
$$
\begin{aligned}
&\int_{K} \nabla (u^2)\cdot \boldsymbol{q} = - \int_{K} u^2 (\mathrm{div}\, \boldsymbol{q}) + \int_{\partial K} u^2 \boldsymbol{q}
\cdot \boldsymbol{n} ds
\\
\implies & \int_{K} 2u\nabla u \cdot \boldsymbol{q} = - \int_{K} u^2 (\mathrm{div}\, \boldsymbol{q})
+ \int_{\partial K} u^2 ds.
\end{aligned}
$$
As a result:
$$
\begin{aligned}
\|{u}\|_{0,\partial K}^2 &= \int_{\partial K} u^2 ds
\lesssim 2 \sup_{K} |\boldsymbol{q}| \int_{K} |u| \,|\nabla u|
+ \sup_{K} |\mathrm{div}\, \boldsymbol{q}| \int_{K} |u|^2
\\
& \lesssim 2 \sup_{K} |\boldsymbol{q}| \,\|{u}_{0,K}\|\, \|{\nabla u}\|_{0,K}
+ \sup_{K} |\mathrm{div}\, \boldsymbol{q}| \, \|{u}\|_{0,K}^2
\\
& \lesssim \sup_{K} \{|\boldsymbol{q}|,|\mathrm{div}\, \boldsymbol{q}| \}\, \|{u}\|_{0,K} \,|u|_{1,K}.
\end{aligned}
$$
By compatibility condition for the Neumann problem $(N)$, $\int_{K} f =
|\partial K|$. This implies that the $\sup_K|f_K| = \sup_K|\mathrm{div}\, \boldsymbol{q}|$ can not be less
than $\frac{|\partial K|}{|K|}$, which is $O(1/h)$ for a shape-regular $K$. By this kind of
argument, the constant is certainly dependent on $h^{-1/2}$ in the trace
inequality $(T)$. However, notice $\sup_{K} |\mathrm{div}\, \boldsymbol{q}|$ is a factor on the
$L^2$-term, which can be bounded with a constant only depending on $|K|^{1/2}$ using
Poincaré-Friedrichs type inequality if $K$ satisfies the extension property. Thus,
the dependence with the domain size is removed.
References
Grisvard, P. (1985) Elliptic Problems in Nonsmooth Domains. SIAM.
Nečas, J. (1967) Les methodes directes en theorie des equations elliptiques. Masson et Cie, Academia.
Nečas, J. (2011) Direct methods in the theory of elliptic equations. Springer Science & Business Media.
Jonsson, A. and Wallin, H. (1978) “A Whitney extension theorem in L^p and Besov spaces,” in Annales de l’institut Fourier, pp. 139–192.
McLean, W. C. H. (2000) Strongly elliptic systems and boundary integral equations. Cambridge university press.
Lions, J. L. and Magenes, E. (2012) Non-homogeneous boundary value problems and applications. Springer Science & Business Media.
Triebel, H. (2010) Theory of Function Spaces. Springer Basel (Modern Birkhäuser Classics).
Marschall, J. (1987) “The trace of Sobolev-Slobodeckij spaces on Lipschitz domains,” manuscripta mathematica. Springer, 58(1), pp. 47–65.
Nikol’skiĭ, S. M., Lizorkin, P. I. and Miroshin, N. V. (1988) “Weighted function spaces and their applications to the investigation of boundary value problems for degenerate elliptic equations,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. Kazan (Volga region) Federal University, (8), pp. 4–30.
Maz’ya, V. G. (1984) “Functions with a finite dirichlet integral in a domain with cusp points on the boundary,” Journal of Soviet Mathematics. Springer, 27(1), pp. 2500–2514.
Yakovlev, G. N. (1967) “Traces of functions in the space W^l,p on piecewise smooth surfaces,” Matematicheskii Sbornik. Russian Academy of Sciences, Branch of Mathematical Sciences, 116(4), pp. 526–543.
Ding, Z. (1996) “A proof of the trace theorem of Sobolev spaces on Lipschitz domains,” Proceedings of the American Mathematical Society, 124(2), pp. 591–600.
Kim, D. (2007) “Trace theorems for Sobolev-Slobodeckij spaces with or without weights,” Journal of Function Spaces. Hindawi Publishing Corporation, 5(3), pp. 243–268.
Arnold, D. N., Scott, L. R. and Vogelius, M. (1988) “Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon,” Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 15(2), pp. 169–192.
Aronszajn, N. (1955) “Boundary values of functions with finite Dirichlet integral,” Techn. Report of Univ. of Kansas, 14, pp. 77–94.
Gagliardo, E. (1957) “Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili,” Rendiconti del seminario matematico della universita di Padova, 27, pp. 284–305.
Slobodeckij, L. N. (1958) “Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations, Leningrad,” Gos. Ped. Inst. Ucep. Zap, 197, pp. 54–112.
Ern, A. and Guermond, J.-L. (2013) Theory and practice of finite elements. Springer Science & Business Media.
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