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diff --git a/_images/01375539e3609d097a9f8c6194ad741f3e45ad6d53d9db36f26811c016b29f56.png b/_images/01375539e3609d097a9f8c6194ad741f3e45ad6d53d9db36f26811c016b29f56.png
diff --git a/_images/2c9d08fc744c61ee49ff74a2fc0ac68791897484e3f4f5e5c9473b2650a4c9d4.png b/_images/2c9d08fc744c61ee49ff74a2fc0ac68791897484e3f4f5e5c9473b2650a4c9d4.png
diff --git a/_images/82b78d013c0f97aedd8d701965430d70885f52f61db66b909cd89aee60142251.png b/_images/82b78d013c0f97aedd8d701965430d70885f52f61db66b909cd89aee60142251.png
diff --git a/_images/Lipschitz.png b/_images/Lipschitz.png
diff --git a/_images/e5a89c290facfa91384e646ee2704313b8901f803fb43a595a4b713a0e106caf.png b/_images/e5a89c290facfa91384e646ee2704313b8901f803fb43a595a4b713a0e106caf.png
diff --git a/_images/extension.png b/_images/extension.png
diff --git a/_images/f2fcc2630a28bb73f11c8355fc2a794632b6a03b27eaf01eb7a461fcb10b2210.png b/_images/f2fcc2630a28bb73f11c8355fc2a794632b6a03b27eaf01eb7a461fcb10b2210.png
diff --git a/_images/tracemapping.png b/_images/tracemapping.png
diff --git a/_sources/sobolevspaces/SobolevSpaces.ipynb b/_sources/sobolevspaces/SobolevSpaces.ipynb
@@ -88,7 +88,11 @@
     "\n",
     "**Theorem:** Let $1 \\leq p < \\infty$. Then $\\widetilde W_p^k  = W_p^k$.\n",
     "\n",
-    "> A domain is called a Lipschitz domain, if its boundary $\\partial \\Omega$ can be locally represented by a Lipschitz-continuous function. \n",
+    "> **Definition:** A domain is called a Lipschitz domain, if its boundary $\\partial \\Omega$ can be locally represented by a Lipschitz-continuous function. \n",
+    "\n",
+    "Are these domains Lipschitz domains?\n",
+    "\n",
+    "<img src=\"Lipschitz.png\" alt=\"Alternative text\" width=\"300\" align=\"center\"/>\n",
     "\n",
     "A precise definition which applies also to unbounded domains is the following:\n",
     "\n",

diff --git a/_sources/sobolevspaces/Traces.ipynb b/_sources/sobolevspaces/Traces.ipynb
@@ -109,6 +109,14 @@
     "\\end{align*}"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "b7449baa-a5e4-4670-8580-c8ce33b83ad4",
+   "metadata": {},
+   "source": [
+    "<img src=\"tracemapping.png\" alt=\"Alternative text\" width=\"300\" align=\"center\"/>"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "e35a0cf2-bc59-4e53-839f-d053be685bd1",
@@ -307,6 +315,9 @@
     "for the  square, and restrict the result onto the general domain $\\Omega$.\n",
     "This is now the motivation to study extension operators.\n",
     "\n",
+    "<img src=\"extension.png\" alt=\"Alternative text\" width=\"300\" align=\"center\"/>\n",
+    "\n",
+    "\n",
     "We construct a non-overlapping covering $\\{ S_i \\}$ of a neighbourhood \n",
     "of $\\partial \\Omega$ on both sides. \n",
     "Let $\\partial \\Omega = \\cup \\Gamma_i$ consist of smooth parts.\n",

diff --git a/_sources/sobolevspaces/exercises.ipynb b/_sources/sobolevspaces/exercises.ipynb
@@ -104,7 +104,7 @@
     "\n",
     "Find all $\\alpha \\in [0,1]$ such that $u \\in H^\\alpha ( (-\\pi, \\pi) )$. \n",
     "\n",
-    "Hint: compute Fourier coefficients $a_i$ and $b_i$ and find all $\\alpha$ such that $\\sum_{i=0}^\\infty i^{2s} (a_i^2 + b_i^2) < \\infty$\n",
+    "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 $\\sum_{k=0}^\\infty k^{2\\alpha} (a_k^2 + b_k^2) < \\infty$\n",
     "\n",
     "### Point evaluation functional\n",
     "\n",
@@ -119,7 +119,8 @@
     "$$\n",
     "f(v) \\leq \\, c \\, \\| v \\|_{H^\\alpha}\n",
     "$$\n",
-    "\n"
+    "\n",
+    "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 $b_k$. "
    ]
   },
   {

diff --git a/_sources/sobolevspaces/experiments.ipynb b/_sources/sobolevspaces/experiments.ipynb
diff --git a/_sources/sobolevspaces/preciseweak.ipynb b/_sources/sobolevspaces/preciseweak.ipynb
@@ -58,12 +58,13 @@
    "id": "b0c7aee2-84ee-4a5f-944f-f3efa7f645d6",
    "metadata": {},
    "source": [
-    "> Theorem: The weak formulation of the Poisson problem: find $u \\in V_D$ such that\n",
+    "> **Theorem:** The weak formulation of the Poisson problem: find $u \\in V_D$ such that\n",
     ">\n",
     "> $$\n",
     "  A(u,v) = f(v) \\qquad \\forall \\, v \\in V_0\n",
     "  $$\n",
-    "> has a unique solution $u$.\n",
+    "> has a unique solution $u \\in V$.\n",
+    "\n",
     "*Proof:* \n",
     "The bilinear-form $A(.,.)$ and the linear-form $f(.)$ are continuous\n",
     "on $V$. Tartar's theorem of equivalent norms proves that $A(.,.)$ is coercive\n",

diff --git a/genindex.html b/genindex.html
@@ -200,7 +200,7 @@
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/Traces.html">15. Trace theorems and their applications</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/equivalentnorms.html">16. Equivalent norms on <span class="math notranslate nohighlight">\(H^1\)</span> and on sub-spaces</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/preciseweak.html">17. The weak formulation of the Poisson equation</a></li>
-<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms (WIP)</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/exercises.html">19. Exercises</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Finite Element Method</span></p>

diff --git a/intro.html b/intro.html
@@ -202,7 +202,7 @@
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/Traces.html">15. Trace theorems and their applications</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/equivalentnorms.html">16. Equivalent norms on <span class="math notranslate nohighlight">\(H^1\)</span> and on sub-spaces</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/preciseweak.html">17. The weak formulation of the Poisson equation</a></li>
-<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms (WIP)</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/exercises.html">19. Exercises</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Finite Element Method</span></p>
@@ -601,7 +601,7 @@ <h2>Installing NGSolve<a class="headerlink" href="#installing-ngsolve" title="Li
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/Traces.html">15. Trace theorems and their applications</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/equivalentnorms.html">16. Equivalent norms on <span class="math notranslate nohighlight">\(H^1\)</span> and on sub-spaces</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/preciseweak.html">17. The weak formulation of the Poisson equation</a></li>
-<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms (WIP)</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/exercises.html">19. Exercises</a></li>
 </ul>
 </div>

diff --git a/mixedelasticity/dynamics.html b/mixedelasticity/dynamics.html
@@ -203,7 +203,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/Traces.html">15. Trace theorems and their applications</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/equivalentnorms.html">16. Equivalent norms on <span class="math notranslate nohighlight">\(H^1\)</span> and on sub-spaces</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/preciseweak.html">17. The weak formulation of the Poisson equation</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/experiments.html">18. Experiments with norms (WIP)</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/experiments.html">18. Experiments with norms</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/exercises.html">19. Exercises</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Finite Element Method</span></p>

diff --git a/objects.inv b/objects.inv
diff --git a/plates/tdnnsplate.html b/plates/tdnnsplate.html
@@ -203,7 +203,7 @@
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/Traces.html">15. Trace theorems and their applications</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/equivalentnorms.html">16. Equivalent norms on <span class="math notranslate nohighlight">\(H^1\)</span> and on sub-spaces</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/preciseweak.html">17. The weak formulation of the Poisson equation</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/experiments.html">18. Experiments with norms (WIP)</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/experiments.html">18. Experiments with norms</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../sobolevspaces/exercises.html">19. Exercises</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Finite Element Method</span></p>

diff --git a/search.html b/search.html
@@ -202,7 +202,7 @@
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/Traces.html">15. Trace theorems and their applications</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/equivalentnorms.html">16. Equivalent norms on <span class="math notranslate nohighlight">\(H^1\)</span> and on sub-spaces</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/preciseweak.html">17. The weak formulation of the Poisson equation</a></li>
-<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms (WIP)</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sobolevspaces/experiments.html">18. Experiments with norms</a></li>
 <li class="toctree-l1"><a class="reference internal" href="sobolevspaces/exercises.html">19. Exercises</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Finite Element Method</span></p>

diff --git a/searchindex.js b/searchindex.js
diff --git a/sobolevspaces/SobolevSpaces.html b/sobolevspaces/SobolevSpaces.html
@@ -530,8 +530,10 @@ <h1><span class="section-number">14. </span>Sobolev spaces<a class="headerlink"
 <p>Under moderate restrictions, these definitions lead to the same spaces:</p>
 <p><strong>Theorem:</strong> Let <span class="math notranslate nohighlight">\(1 \leq p &lt; \infty\)</span>. Then <span class="math notranslate nohighlight">\(\widetilde W_p^k  = W_p^k\)</span>.</p>
 <blockquote>
-<div><p>A domain is called a Lipschitz domain, if its boundary <span class="math notranslate nohighlight">\(\partial \Omega\)</span> can be locally represented by a Lipschitz-continuous function.</p>
+<div><p><strong>Definition:</strong> A domain is called a Lipschitz domain, if its boundary <span class="math notranslate nohighlight">\(\partial \Omega\)</span> can be locally represented by a Lipschitz-continuous function.</p>
 </div></blockquote>
+<p>Are these domains Lipschitz domains?</p>
+<a class="reference internal image-reference" href="../_images/Lipschitz.png"><img alt="Alternative text" class="align-center" src="../_images/Lipschitz.png" style="width: 300px;" /></a>
 <p>A precise definition which applies also to unbounded domains is the following:</p>
 <p><strong>Definition:</strong> The domain <span class="math notranslate nohighlight">\(\Omega\)</span> has a <strong>Lipschitz boundary</strong>, <span class="math notranslate nohighlight">\(\partial \Omega\)</span>, if there exists a collection of open sets <span class="math notranslate nohighlight">\(O_i\)</span>, a positive parameter <span class="math notranslate nohighlight">\(\varepsilon\)</span>, an integer <span class="math notranslate nohighlight">\(N\)</span> and a finite number <span class="math notranslate nohighlight">\(L\)</span>, such that for all <span class="math notranslate nohighlight">\(x \in \partial \Omega\)</span> the ball of radius <span class="math notranslate nohighlight">\(\varepsilon\)</span> centered at <span class="math notranslate nohighlight">\(x\)</span> is contained in some <span class="math notranslate nohighlight">\(O_i\)</span>, no more than <span class="math notranslate nohighlight">\(N\)</span> of the sets <span class="math notranslate nohighlight">\(O_i\)</span> intersect non-trivially, and each
 part of the boundary <span class="math notranslate nohighlight">\(O_i \cap \Omega\)</span> is a graph of a Lipschitz function <span class="math notranslate nohighlight">\(\varphi_i : {\mathbb R}^{d-1} \rightarrow {\mathbb R}\)</span> with Lipschitz norm bounded by <span class="math notranslate nohighlight">\(L\)</span>.</p>

diff --git a/sobolevspaces/Traces.html b/sobolevspaces/Traces.html
@@ -579,6 +579,7 @@ <h1><span class="section-number">15. </span>Trace theorems and their application
  c \, \int_0^1  \int_0^1  \left\{ u(\xi,\eta)^2 + \left(\frac{\partial u}{\partial \eta}(\xi,\eta) \right)^2 \right\} \, d \eta \, d \xi \\
         &amp; \leq  c \, \| u \|_{H^1(Q)}^2.
 \end{align*}\]</div>
+<a class="reference internal image-reference" href="../_images/tracemapping.png"><img alt="Alternative text" class="align-center" src="../_images/tracemapping.png" style="width: 300px;" /></a>
 <p>We use the partitioning of <span class="math notranslate nohighlight">\(\partial \Omega\)</span> into non-overlapping parts <span class="math notranslate nohighlight">\(\Gamma_i\)</span>, <span class="math notranslate nohighlight">\(1 \leq i \leq M\)</span>. For each part <span class="math notranslate nohighlight">\(\Gamma_i\)</span> we define an injective transformation <span class="math notranslate nohighlight">\(s_i : \overline Q \rightarrow \overline \Omega\)</span> such that the bottom edge of <span class="math notranslate nohighlight">\(Q\)</span> is mapped onto <span class="math notranslate nohighlight">\(\Gamma_i\)</span>. We define <span class="math notranslate nohighlight">\(S_i = s_i(Q)\)</span>. We assume that <span class="math notranslate nohighlight">\(s_i^\prime\)</span> and <span class="math notranslate nohighlight">\((s_i^\prime)^{-1}\)</span> are bounded functions a.e.</p>
 <p>This allows to pull-back functions <span class="math notranslate nohighlight">\(u \in H^1(S_i)\)</span> locally to the square domain <span class="math notranslate nohighlight">\(Q\)</span>:</p>
 <div class="math notranslate nohighlight">
@@ -730,6 +731,7 @@ <h2><span class="section-number">15.3. </span>Extension operators<a class="heade
 a function <span class="math notranslate nohighlight">\(u \in H^1(\Omega)\)</span> onto a larger square <span class="math notranslate nohighlight">\(Q\)</span>, apply the result
 for the  square, and restrict the result onto the general domain <span class="math notranslate nohighlight">\(\Omega\)</span>.
 This is now the motivation to study extension operators.</p>
+<a class="reference internal image-reference" href="../_images/extension.png"><img alt="Alternative text" class="align-center" src="../_images/extension.png" style="width: 300px;" /></a>
 <p>We construct a non-overlapping covering <span class="math notranslate nohighlight">\(\{ S_i \}\)</span> of a neighbourhood
 of <span class="math notranslate nohighlight">\(\partial \Omega\)</span> on both sides.
 Let <span class="math notranslate nohighlight">\(\partial \Omega = \cup \Gamma_i\)</span> consist of smooth parts.
@@ -762,8 +764,8 @@ <h2><span class="section-number">15.3. </span>Extension operators<a class="heade
 \]</div>
 <p>Define the domain <span class="math notranslate nohighlight">\(\widetilde \Omega = \Omega \cup S_1 \cup \ldots \cup S_M\)</span>.</p>
 <p>We define the extension operator by</p>
-<div class="amsmath math notranslate nohighlight" id="equation-e42f7476-fef8-404f-83e4-49890ef6ed4f">
-<span class="eqno">()<a class="headerlink" href="#equation-e42f7476-fef8-404f-83e4-49890ef6ed4f" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-db688cc6-56eb-421c-980a-3f3724546b80">
+<span class="eqno">()<a class="headerlink" href="#equation-db688cc6-56eb-421c-980a-3f3724546b80" title="Permalink to this equation">#</a></span>\[\begin{equation}
 \begin{array}{rcll}
 (E u) (\hat x) &amp; = &amp; u(x) &amp; \qquad \forall \, x \in \cup S_i \\
 (Eu) (x) &amp; = &amp; u(x) &amp; \qquad \forall \, x \in \Omega
@@ -998,8 +1000,8 @@ <h3><span class="section-number">15.5.1. </span>General definition:<a class="hea
 \]</div>
 <p>By definition of the weak derivative, there holds <span class="math notranslate nohighlight">\((z_k^\prime)^\prime = (1-\lambda_k) z_k\)</span>, i.e., <span class="math notranslate nohighlight">\(z^k \in H^2\)</span>. Since <span class="math notranslate nohighlight">\(H^2 \subset C^0\)</span>, there holds also
 <span class="math notranslate nohighlight">\(z \in C^2\)</span>, and a weak solution is also a solution of the strong form</p>
-<div class="amsmath math notranslate nohighlight" id="equation-a9a86b0a-3322-4b4f-aba4-02c90aa2d0bb">
-<span class="eqno">()<a class="headerlink" href="#equation-a9a86b0a-3322-4b4f-aba4-02c90aa2d0bb" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-d4c7c393-7e43-4018-abf4-2b99875187d1">
+<span class="eqno">()<a class="headerlink" href="#equation-d4c7c393-7e43-4018-abf4-2b99875187d1" title="Permalink to this equation">#</a></span>\[\begin{equation}
 \begin{array}{rcll}
 z_k - z_k^{\prime \prime} &amp; = &amp; \lambda_k z_k \qquad &amp; \mbox{ on } (0,1) \\
 z_k^\prime(0) = z_k^\prime(1) &amp; = &amp; 0

diff --git a/sobolevspaces/exercises.html b/sobolevspaces/exercises.html
@@ -572,7 +572,7 @@ <h3><span class="section-number">19.4.1. </span>Step function<a class="headerlin
       \end{array} \right.
 \end{split}\]</div>
 <p>Find all <span class="math notranslate nohighlight">\(\alpha \in [0,1]\)</span> such that <span class="math notranslate nohighlight">\(u \in H^\alpha ( (-\pi, \pi) )\)</span>.</p>
-<p>Hint: compute Fourier coefficients <span class="math notranslate nohighlight">\(a_i\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> and find all <span class="math notranslate nohighlight">\(\alpha\)</span> such that <span class="math notranslate nohighlight">\(\sum_{i=0}^\infty i^{2s} (a_i^2 + b_i^2) &lt; \infty\)</span></p>
+<p>Hint: compute Fourier coefficients <span class="math notranslate nohighlight">\(a_k\)</span> and <span class="math notranslate nohighlight">\(b_k\)</span> of <span class="math notranslate nohighlight">\(u(x) = \sum a_k \cos(k x) + b_k \sin(k x)\)</span>, and find all <span class="math notranslate nohighlight">\(\alpha\)</span> such that <span class="math notranslate nohighlight">\(\sum_{k=0}^\infty k^{2\alpha} (a_k^2 + b_k^2) &lt; \infty\)</span></p>
 </section>
 <section id="point-evaluation-functional">
 <h3><span class="section-number">19.4.2. </span>Point evaluation functional<a class="headerlink" href="#point-evaluation-functional" title="Link to this heading">#</a></h3>
@@ -586,6 +586,7 @@ <h3><span class="section-number">19.4.2. </span>Point evaluation functional<a cl
 \[
 f(v) \leq \, c \, \| v \|_{H^\alpha}
 \]</div>
+<p>Hint: Expand <span class="math notranslate nohighlight">\(v\)</span> as a Fourier series <span class="math notranslate nohighlight">\(v(x) = \sum a_k \cos(k x) + b_k \sin(k x)\)</span>, and express <span class="math notranslate nohighlight">\(f(v)\)</span> by means of the <span class="math notranslate nohighlight">\(a_k\)</span> and <span class="math notranslate nohighlight">\(b_k\)</span>.</p>
 </section>
 </section>
 </section>

diff --git a/sobolevspaces/experiments.html b/sobolevspaces/experiments.html