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JSchoeberl committed Mar 8, 2024
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4 changes: 2 additions & 2 deletions _sources/abstracttheory/exercises.ipynb
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"$$\n",
"\n",
"Show that the bilinear-form is continuous.\n",
"Proof that the bilinear-form satisfies both $\\inf-\\sup$ conditions, start with\n",
"Prove that the bilinear-form satisfies both $\\inf-\\sup$ conditions, start with\n",
"\n",
"$$\n",
"\\inf_{ u \\in V} \\sup_{v \\in W} \\frac{B(u,v)}{ \\| u \\|_V \\| v \\|_W} \\geq \\beta_1 > 0\n",
"$$\n",
"\n",
"For a given $u \\in V$, find some $v \\in W$ such that the fraction is large (how ?). This proves that the derivative is one-to-one on $V$. What if the boundary condition $v(0)=0$ is skipped ? \n",
"\n",
"When swap order of the $\\inf-\\sup$ condition:\n",
"Then swap order of the $\\inf-\\sup$ condition:\n",
"\n",
"$$\n",
"\\inf_{ v \\in W} \\sup_{u \\in V} \\frac{B(u,v)}{ \\| u \\|_V \\| v \\|_W} \\geq \\beta_1 > 0\n",
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6 changes: 3 additions & 3 deletions abstracttheory/exercises.html
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B(u,v) = \int_0^1 u^\prime v \, dx
\]</div>
<p>Show that the bilinear-form is continuous.
Proof that the bilinear-form satisfies both <span class="math notranslate nohighlight">\(\inf-\sup\)</span> conditions, start with</p>
Prove that the bilinear-form satisfies both <span class="math notranslate nohighlight">\(\inf-\sup\)</span> conditions, start with</p>
<div class="math notranslate nohighlight">
\[
\inf_{ u \in V} \sup_{v \in W} \frac{B(u,v)}{ \| u \|_V \| v \|_W} \geq \beta_1 &gt; 0
\]</div>
<p>For a given <span class="math notranslate nohighlight">\(u \in V\)</span>, find some <span class="math notranslate nohighlight">\(v \in W\)</span> such that the fraction is large (how ?). This proves that the derivative is one-to-one on <span class="math notranslate nohighlight">\(V\)</span>. What if the boundary condition <span class="math notranslate nohighlight">\(v(0)=0\)</span> is skipped ?</p>
<p>When swap order of the <span class="math notranslate nohighlight">\(\inf-\sup\)</span> condition:</p>
<p>Then swap order of the <span class="math notranslate nohighlight">\(\inf-\sup\)</span> condition:</p>
<div class="math notranslate nohighlight">
\[
\inf_{ v \in W} \sup_{u \in V} \frac{B(u,v)}{ \| u \|_V \| v \|_W} \geq \beta_1 &gt; 0
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