19. Exercises#

19.1. Friedrichs’ inequality#

19.1.1. Friedrichs’ inequality in 1D#

Prove Friedrichs’ inequality on the interval \(I = (a,b)\) with elementary tools. Show that there exists a constant \(c_F\) such that

\[ \| u \|_{L_2(I)} \leq c_F \, \| u^\prime \|_{L_2(I)} \qquad \forall \, u \in C^1(\overline{I}), \, u(a) = 0 \]

how does your constant \(c_F\) depend on \(a\) and \(b\) ?

19.1.2. Friedrichs’ inequality on the square#

Define \(\Omega = (0,a) \times (0,b)\). Prove Friedrichs’ inequality

\[ \| u \|_{L_2(\Omega)} \leq c_F \| \nabla u \|_{L_2(\Omega)} \qquad \forall \, u \in C^1(\overline\Omega), \; u(0,y) = 0 \text{ for } 0 \leq y \leq b. \]

How does \(c_F\) depend on \(a\) and \(b\) ?

19.2. Poincaré inequality#

19.2.1. Eqivalent versions of the Poincaré inequality#

Prove that the following inequalities are equivalent

  1. There exists a constant \(c_p\) such that

\[ \| u - \overline u \|_{L_2}^2 \leq c_P^2 \, \| \nabla u \|^2_{L_2} \; \forall \, u \in H^1(\Omega)\]

Here, \(\overline u\) is the mean value of \(u\), which is \(\frac{1}{|\Omega|} \int_\Omega u(x) dx\), understood as a constant function.

  1. There exists a constant \(c_p\) such that

    \[ \| u \|_{L_2}^2 \leq c_P^2 \, \| \nabla u \|_{L_2}^2 + \frac{1}{|\Omega|} \left( \int_\Omega u \right)^2 \]

Hint: Pythagoras. First, observe that \(\overline u\) and \(u - \overline u\) are orthogonal with respect to the \(L_2\) inner product.

19.2.2. Poincaré inequality in 1D#

Prove the Poincaré inequality on the interval \(I = (a,b)\) with elementary tools:

\[ \| u - \overline u \|_{L_2(I)}^2 \leq c_P^2 \| u^\prime \|_{L_2(I)}^2 \qquad \forall \, u \in C^1(\overline I) \]

How does the Poincaré constant depend on \(a\) and \(b\) ?

Hint: Bring the left hand side to something similar to \(\int \left( \int u(y) - u(x) dx\right) ^2 dy\). Then, use the fundamental theorem of calculus: \(u(y)-u(x) = \int_x^y u^\prime(s) \, ds\). finally: Cauchy-Schwarz

19.3. Bramble-Hilbert Lemma#

19.3.1. Mean-value interpolation#

Let \(\Omega \subset {\mathbb R}^d\), open, bounded and connected. Prove that there exists a constant \(c\) such that

\[ \| u - \overline u \|_{L_2(\Omega)} \leq c \, \| \nabla u \|_{L_2(\Omega)} \qquad \forall \, u \in H^1(\Omega), \]

where \(\overline u := | \Omega |^{-1} \int_\Omega u \, dx\) is the mean value.

Use the Bramble-Hilbert theorem

19.3.2. Scaled domain#

Let \(\Omega = B_r(p)\) a ball with center \(p\) and radius \(r\). Prove that there exists a constant \(c\), independent of \(r\) and \(p\), such that

\[ \| u - \overline u \|_{L_2} \leq c r \, \| \nabla u \|_{L_2} \qquad \forall \, u \in H^1(B_r(p)) \]

Hint: Prove the estimate for the unit ball \(B_1(0)\). Define a function \(\Phi\) mapping the unit-ball to the arbitrary ball \(B_r(p)\). For \(u \in H^1(B_r(p))\), define the pull-back \(u \circ \Phi \in H^1(B_1(0))\). Does mean-value and pull-back commute, i.e. does there hold \(\overline u \circ \Phi = \overline{ u \circ \Phi}\) ?

19.4. Fractional Sobolev spaces#

19.4.1. Step function#

Consider the function \(u : (-\pi, \pi) \rightarrow {\mathbb R}\) defined as

\[\begin{split} u(x) = \left\{ \begin{array}{cl} -1 & x < 0 \\ 1 & x \geq 0 \end{array} \right. \end{split}\]

Find all \(\alpha \in [0,1]\) such that \(u \in H^\alpha ( (-\pi, \pi) )\).

Hint: compute Fourier coefficients \(a_k\) and \(b_k\) of \(u(x) = \sum a_k \cos(k x) + b_k \sin(k x)\), and find all \(\alpha\) such that \(\sum_{k=0}^\infty k^{2\alpha} (a_k^2 + b_k^2) < \infty\)

19.4.2. Point evaluation functional#

Consider the linear functional

\[ f : C([-\pi,\pi]) \rightarrow {\mathbb R} : v \mapsto v(0) \]

Find all \(\alpha \in (0,1)\) such that \(f\) can be continuously extended onto \(H^\alpha((-\pi,\pi))\), i.e. find all \(\alpha\) such that

\[ f(v) \leq \, c \, \| v \|_{H^\alpha} \]

Hint: Expand \(v\) as a Fourier series \(v(x) = \sum a_k \cos(k x) + b_k \sin(k x)\), and express \(f(v)\) by means of the \(a_k\) and \(b_k\).