16. Equivalent norms on \(H^1\) and on sub-spaces

The intention is to formulate \(2^{nd}\) order variational problems in the Hilbert space \(H^1\). We want to apply the Lax-Milgram theory for continuous and coercive bilinear forms \(A(.,.)\). We present techniques to prove coercivity.

The idea is the following. In the norm

\[ \| v \|_{H^1}^2 = \| v \|_{L_2}^2 + \| \nabla v \|_{L_2}^2, \]

the \(\| \nabla \cdot \|_{L_2}\)-semi-norm is the dominating part up to the constant functions. The \(L_2\) norm is necessary to obtain a norm. We want to replace the \(L_2\) norm by some different term (e.g., the \(L_2\)-norm on a part of \(\Omega\), or the \(L_2\)-norm on \(\partial \Omega\)), and want to obtain an equivalent norm.

We formulate an abstract theorem relating a norm \(\|.\|_V\) to a semi-norm \(\|.\|_A\). This theorem can be found as Thm 2.1 in the textbook Finite Element Methods for Naiver-Stokes Equations by Girault and Raviart:

Theorem by Tartar:

Let \((V, (.,.)_V)\) and \((W, (.,.)_W)\) be Hilbert spaces, such that the embedding \(id : V \rightarrow W\) is compact. Let \(A(.,.)\) be a non-negative, symmetric and \(V\)-continuous bilinear form with kernel \(V_0 = \{ v : A(v,v) = 0 \}\). Assume that

\[ \| v \|_V^2 \simeq \| v \|_W^2 + \| v \|_A^2 \qquad \forall \, v \in V \]

Then there holds

1. The kernel \(V_0\) is finite dimensional. On the factor space \(V/V_0\), \(A(.,.)\) is an equivalent norm to the quotient norm

\[ \| u \|_A \simeq \inf_{v \in V_0} \| u - v \|_V \qquad \forall \, u \in V \]

2. Let \(B(.,.)\) be a continuous, non-negative, symmetric bilinear form on \(V\) such that \(A(.,.) + B(.,.)\) is an inner product. Then there holds

\[ \| v \|_V^2 \simeq \| v \|_A^2 + \| v \|_B^2 \qquad \forall \, v \in V \]

3. Let \(V_1 \subset V\) be a closed sub-space such that \(V_0 \cap V_1 = \{ 0 \}\). Then there holds

\[ \| v \|_V \simeq \| v \|_A \qquad \forall \, v \in V_1 \]

Proof: 1. Assume that \(V_0\) is not finite dimensional. Then there exists an \((.,.)_V\)-orthonormal sequence in \(V_0\). Since the embedding \(id : V \rightarrow W\) is compact, it has a sub-sequence named \((u_k)\) converging in \(\|. \|_W\). But, since

\[ 2 = \| u_k - u_l \|_V^2 \simeq \| u_k - u_l \|_W^2 + \| u_k - u_l \|_A = \| u_k - u_l \|_W^2 \]

for \(k \neq l\), \(u_k\) is not Cauchy in \(W\). This is a contradiction to an infinite dimensional kernel space \(V_0\). We prove the equivalence to the quotient norm. To bound the left hand side by the right hand side, we use that \(V_0 = \operatorname {ker} A\), and norm equivalence assumption:

\[ \| u \|_A = \inf_{v \in V_0} \| u - v \|_A \leq \inf_{v \in V_0} \| u - v \|_V \]

The quotient norm is equal to \(\| P_{V_0^\bot} u \|_V\). We have to prove that \(\| P_{V_0^\bot} u \|_V \leq \| P_{V_0^\bot} u \|_A\) for all \(u \in V\). This follows after proving \(\| u \|_V \leq \| u \|_A\) for all \(u \in V_0^\bot\). Assume that this is not true. I.e., there exists a \(V\)-orthogonal sequence \((u_k)\) such that \(\| u_k \|_A \leq k^{-1} \| u_k \|_V\). Extract a sub-sequence converging in \(\| . \|_W\), and call it \(u_k\) again. From the norm equivalence in the assumption there follows

\[ 2 = \| u_k - u_l \|_V^2 \preceq \| u_k - u_l \|_W + \| u_k - u_l \|_A \rightarrow 0 \]

2. On \(V_0\), \(\| . \|_B\) is a norm. Since \(V_0\) is finite dimensional, it is equivalent to \(\|.\|_V\), say with bounds

\[ c_1 \, \| v \|_{V}^2 \leq \| v \|_B^2 \leq c_2 \, \| v \|_V^2 \qquad \forall \, v \in V_0 \]

From 1. we know that

\[ c_3 \, \| v \|_V^2 \leq \| v \|_A^2 \leq c_4 \, \| v \|_V^2 \qquad \forall \, v \in V_0^\bot. \]

Now, we bound

\[\begin{align*} \| u \|_V^2 & = \| P_{V_0} u \|_V^2 + \| P_{V_0^\bot} u \|_V^2 \\ & \leq \frac{1}{c_1} \| \underbrace{P_{V_0} u }_{u - P_{V_0^\bot} u } \|_B^2 + \| P_{V_0^\bot} \|_V^2 \\ & \leq \frac{2}{c_1} \left( \| u \|_B^2 + c_2 \| P_{V_0^\bot} u \|_V^2 \right) + \| P_{V_0^\bot} u \|_V^2 \\ & = \frac{2}{c_1} \| u \|_B^2 + \frac{1}{c_2} \left(1 + \frac{2 c_2}{c_1} \right) \| P_{V_0^\bot} u \|_A^2 \\ & \preceq \| u \|_B^2 + \| u \|_A^2 \end{align*}\]

3. Define \(B(u,v) = (P_{V_1}^\bot u, P_{V_1}^\bot u)_V\). Then \(A(.,.)+B(.,.)\) is an inner product: \(A(u,u)+B(u,u) = 0\) implies that \(u \in V_0\) and \(u \in V_1\), thus \(u = \{ 0 \}\). From 2. there follows that \(A(.,.)+B(.,.)\) is equivalent to \((.,.)_V\). The result follows from reducing the equivalence to \(V_1\).

\(\Box\)

We want to apply Tartar’s theorem to the case \(V = H^1\), \(W = L_2\), and \(\|v \|_A = \| \nabla v \|_{L_2}\). The theorem requires that the embedding \(id : H^1 \rightarrow L_2\) is compact. This is indeed true for bounded domains \(\Omega\):

Theorem: The embedding of \(H^k \rightarrow H^l\) for \(k > l\) is compact.

We sketch a proof for the embedding \(H^1 \subset L_2\). First, prove the compact embedding \(H_0^1(Q) \rightarrow L_2(Q)\) for a square \(Q\), w.l.o.g. set \(Q = (0,1)^2\). The eigen-value problem: Find \(z \in H_0^1(Q)\) and \(\lambda\) such that

\[ (z,v)_{L_2} + (\nabla z, \nabla v)_{L_2} = \lambda (u,v)_{L_2} \qquad \forall \, v \in H_0^1(Q) \]

has eigen-vectors \(z_{k,l} = sin (k \pi x) sin (l \pi y)\), and eigen-values \(1+k^2 \pi^2 + l^2 \pi^2 \rightarrow \infty\). The eigen-vectors are dense in \(L_2\). Thus, the embedding is compact.

On a general domain \(\Omega \subset Q\), we can extend \(H^1(\Omega)\) into \(H_0^1(Q)\), embed \(H_0^1(Q)\) into \(L_2(Q)\), and restrict \(L_2(Q)\) onto \(L_2(\Omega)\). This is the composite of two continuous and a compact mapping, and thus is compact. \(\Box\)

The kernel \(V_0\) of the semi-norm \(\| \nabla v \|\) is the 1-dimensional space of all constant functions.

Friedrichs’ inequality: Let \(\Gamma_D \subset \partial \Omega\) be of positive measure \(|\Gamma_D|\). Let \(V_D = \{ v \in H^1(\Omega) : \operatorname{tr}_{\Gamma_D} v = 0 \}\). Then

\[ \| v \|_{L_2} \preceq \| \nabla v \|_{L_2} \qquad \forall \, v \in V_D \]

Proof: The intersection \(V_0 \cap V_D\) is trivial \(\{ 0 \}\). Thus, Tartar’s theorem, part 3, implies the equivalence

\[ \| v \|_V^2 = \| v \|_{L_2}^2 + \| \nabla v \|_{L_2}^2 \simeq \| \nabla v \|_{L_2}^2. \]

\(\Box\)

Poincaré inequality:

There holds

\[ \| v \|_{H^1(\Omega)}^2 \preceq \| \nabla v \|_{L_2(\Omega)}^2 + \left(\int_\Omega v \, dx\right)^2 \]

Proof: \(B(u,v) := (\int_\Omega u \, dx) (\int_\Omega v \, dx)\) is a continuous bilinear form on \(H^1\), and \((\nabla u, \nabla v) + B(u,v)\) is an inner product. Thus, Tartar’s theorem, part 2, implies the stated equivalence. \(\Box\)

Some more immediate corollaries from Tartar’s theorem:

• Let \(\omega \subset \Omega\) have positive measure \(| \omega|\) in \({\mathbb R}^d\). Then

  \[ \| u \|_{H^1(\Omega)}^2 \simeq \| \nabla v \|_{L_2(\Omega)}^2 + \| v \|_{L_2(\omega)}^2, \]

• Let \(\gamma \subset \partial \Omega\) have positive measure \(| \gamma|\) in \({\mathbb R}^{d-1}\). Then

  \[ \| u \|_{H^1(\Omega)}^2 \simeq \| \nabla v \|_{L_2(\Omega)}^2 + \| v \|_{L_2(\gamma)}^2, \]

Bramble Hilbert lemma:

Let \(U\) be some Hilbert space, and \(L : H^k \rightarrow U\) be a continuous linear operator, i.e.

\[ \| L v \|_U \preceq \| v \|_{H^k} \qquad \forall \, v \in H^k. \]

Let \(L\) be such that \(L q = 0\) for all polynomials \(q \in P^{k-1}\). Then one can replace the norm by the semi-norm on the right hand side, i.e. there holds

\[ \| L v \|_U \preceq | v |_{H^k} \qquad \forall \, v \in H^k. \]

Proof: The embedding \(H^k \rightarrow H^{k-1}\) is compact.

The \(V\)-continuous, symmetric and non-negative bilinear form \(A(u,v) = \sum_{\alpha : | \alpha | = k} (\partial^\alpha u, \partial^\alpha v)\) has the kernel \(V_0 = P^{k-1}\). Decompose \(\| u \|_{H^k}^2 = \| u \|_{H^{k-1}}^2 + A(u,u)\). By Tartar’s theorem, part 1, there holds

\[ \| u \|_A \simeq \inf_{v \in V_0} \| u - v \|_{H^k} \qquad \forall \, u \in V. \]

The same holds for the bilinear-form

\[ A_2(u,v) := (L u, Lv)_U + A(u,v) \]

Thus

\[ \| u \|_{A_2} \simeq \inf_{v \in V_0} \| u - v \|_{H^k} \qquad \forall \, u \in V \]

Equalizing both implies that

\[ (L u, Lu)_U \leq \| u \|_{A_2}^2 \simeq \| u \|_A^2 \qquad \forall \, u \in V, \]

i.e., the claim.

We will need point evaluation of functions in Sobolev spaces \(H^s\). This is possible, we \(u \in H^s\) implies that \(u\) is continuous.

Sobolev’s embedding theorem: Let \(\Omega \subset {\mathbb R}^d\) with > Lipschitz boundary. If \(u \in H^s\) with \(s > d/2\), then \(u \in L_\infty\) with

\[ \| u \|_{L_\infty} \preceq \| u \|_{H^s} \]

There is a function in \(C^0\) within the \(L_\infty\) equivalence class.