https://web.ma.utexas.edu/mediawiki/index.php?action=history&feed=atom&title=Free_boundary_problemsFree boundary problems - Revision history2024-03-28T10:21:01ZRevision history for this page on the wikiMediaWiki 1.40.1https://web.ma.utexas.edu/mediawiki/index.php?title=Free_boundary_problems&diff=374&oldid=previmported>Luis: /* Fractional obstacle problem */2015-04-16T02:36:28Z<p><span dir="auto"><span class="autocomment">Fractional obstacle problem</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:36, 15 April 2015</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the <del style="font-weight: bold; text-decoration: none;">[[</del>Almgren frequency formula<del style="font-weight: bold; text-decoration: none;">]]</del>.<ref name="S"/><ref name="CSS"/></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the <ins style="font-weight: bold; text-decoration: none;">limit of the </ins>Almgren frequency formula.<ref name="S"/><ref name="CSS"/></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional Alt-Caffarelli problem]]==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional Alt-Caffarelli problem]]==</div></td></tr>
</table>imported>Luishttps://web.ma.utexas.edu/mediawiki/index.php?title=Free_boundary_problems&diff=373&oldid=previmported>Luis at 06:10, 8 June 20112011-06-08T06:10:31Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:10, 8 June 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Studied problems for $(-\Delta)^s$=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Studied problems for $(-\Delta)^s$=</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Here are some free boundary problems for the fractional <del style="font-weight: bold; text-decoration: none;">Laplace </del>operator $(-\Delta)^s$ that have been studied to some extent. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Here are some free boundary problems for the <ins style="font-weight: bold; text-decoration: none;">[[</ins>fractional <ins style="font-weight: bold; text-decoration: none;">Laplacian]] </ins>operator $(-\Delta)^s$ that have been studied to some extent. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[extension technique]],<ref name="CS"/> representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[extension technique]],<ref name="CS"/> representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below.</div></td></tr>
</table>imported>Luishttps://web.ma.utexas.edu/mediawiki/index.php?title=Free_boundary_problems&diff=372&oldid=previmported>RayAYang: refs2011-05-31T16:38:56Z<p>refs</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:38, 31 May 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2">Line 2:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here are some free boundary problems for the fractional Laplace operator $(-\Delta)^s$ that have been studied to some extent. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here are some free boundary problems for the fractional Laplace operator $(-\Delta)^s$ that have been studied to some extent. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The [[extension technique]], representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The [[extension technique]],<ins style="font-weight: bold; text-decoration: none;"><ref name="CS"/> </ins>representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional obstacle problem]]==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional obstacle problem]]==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].<ins style="font-weight: bold; text-decoration: none;"><ref name="S"/><ref name="CSS"/></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional Alt-Caffarelli problem]]==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional Alt-Caffarelli problem]]==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The problem is to seek the minimizer of an energy which is the sum of the Dirichlet form <del style="font-weight: bold; text-decoration: none;"> </del>corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The problem is to seek the minimizer of an energy which is the sum of the <ins style="font-weight: bold; text-decoration: none;">[[</ins>Dirichlet form<ins style="font-weight: bold; text-decoration: none;">]] </ins>corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Like its 2nd order analogue, solutions to problem are known to have optimal regularity of Holder class $C^s$, and to be $s$-harmonic away from the zero set. They have nondegeneracy of order $s$ as well, which is to say that </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Like its 2nd order analogue, solutions to problem are known to have optimal regularity of Holder class $C^s$, and to be $s$-harmonic away from the zero set. They have nondegeneracy of order $s$ as well, which is to say that </div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where $\Lambda$ is the zero set of the $u$. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where $\Lambda$ is the zero set of the $u$. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat.<ins style="font-weight: bold; text-decoration: none;"><ref name="CRS"/><ref name="DR"/></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>==[[<del style="font-weight: bold; text-decoration: none;">Fractional Alt</del>-<del style="font-weight: bold; text-decoration: none;">Phillips </del>problem]]==</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=<ins style="font-weight: bold; text-decoration: none;">=References==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{reflist|refs=</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal</ins>=[[<ins style="font-weight: bold; text-decoration: none;">Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=An extension problem related to the fractional Laplacian | url=http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 | doi=10.1080/03605300600987306 | year=2007 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=32 | issue=7 | pages=1245–1260}}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref name="CRS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Roquejoffre | first2=Jean-Michel | last3=Sire | first3=Yannick | title=Variational problems for free boundaries for the fractional Laplacian | url=http://dx.doi.org/10.4171/JEMS/226 | doi=10.4171/JEMS/226 | year=2010 | journal=Journal of the European Mathematical Society (JEMS) | issn=1435</ins>-<ins style="font-weight: bold; text-decoration: none;">9855 | volume=12 | issue=5 | pages=1151–1179}}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref name="DR">{{Citation | last1=De Silva | first1=Daniela | last2=Roquejoffre | first2=Jean-Michel | title=Regularity in a one-phase free boundary </ins>problem <ins style="font-weight: bold; text-decoration: none;">for the fractional Laplacian | url=http://arxiv.org/abs/1102.3086v1 | year=2011}}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics</ins>]] <ins style="font-weight: bold; text-decoration: none;">| issn=0010-3640 | volume=60 | issue</ins>=<ins style="font-weight: bold; text-decoration: none;">1 | pages</ins>=<ins style="font-weight: bold; text-decoration: none;">67–112}}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">}}</ins></div></td></tr>
</table>imported>RayAYanghttps://web.ma.utexas.edu/mediawiki/index.php?title=Free_boundary_problems&diff=371&oldid=previmported>RayAYang: /* Fractional obstacle problem */2011-05-24T03:13:56Z<p><span dir="auto"><span class="autocomment">Fractional obstacle problem</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:13, 23 May 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional obstacle problem]]==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional obstacle problem]]==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty<ins style="font-weight: bold; text-decoration: none;">$ if $\Omega$ is, say, all of $\mathbb{R}^n</ins>$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].</div></td></tr>
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</table>imported>RayAYanghttps://web.ma.utexas.edu/mediawiki/index.php?title=Free_boundary_problems&diff=370&oldid=previmported>RayAYang: /* Fractional obstacle problem */2011-05-24T03:13:08Z<p><span dir="auto"><span class="autocomment">Fractional obstacle problem</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:13, 23 May 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional obstacle problem]]==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional obstacle problem]]==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$. The problem can be formulated as a variational problem as well, either through the extension or through a <del style="font-weight: bold; text-decoration: none;">[[Dirchlet form]] </del>on $\mathbb{R}^n$. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$. The problem can be formulated as a variational problem as well, either through the extension or <ins style="font-weight: bold; text-decoration: none;">directly </ins>through a <ins style="font-weight: bold; text-decoration: none;">Dirichlet-like nonlocal energy </ins>on $\mathbb{R}^n$. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]]. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional Alt-Caffarelli problem]]==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Fractional Alt-Caffarelli problem]]==</div></td></tr>
</table>imported>RayAYanghttps://web.ma.utexas.edu/mediawiki/index.php?title=Free_boundary_problems&diff=369&oldid=previmported>RayAYang at 02:43, 24 May 20112011-05-24T02:43:08Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:43, 23 May 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[extension technique]], representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[extension technique]], representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[[Fractional obstacle problem]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==</ins>[[Fractional obstacle problem]]<ins style="font-weight: bold; text-decoration: none;">==</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$. The problem can be formulated as a variational problem as well, either through the extension or through a [[Dirchlet form]] on $\mathbb{R}^n$. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$. The problem can be formulated as a variational problem as well, either through the extension or through a [[Dirchlet form]] on $\mathbb{R}^n$. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]]. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]]. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[[Fractional Alt-Caffarelli problem]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==</ins>[[Fractional Alt-Caffarelli problem]]<ins style="font-weight: bold; text-decoration: none;">==</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The problem is to seek the minimizer of an energy which is the sum of the Dirichlet form corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The problem is to seek the minimizer of an energy which is the sum of the Dirichlet form corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l18">Line 18:</td>
<td colspan="2" class="diff-lineno">Line 18:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[[Fractional Alt-Phillips problem]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==</ins>[[Fractional Alt-Phillips problem]]<ins style="font-weight: bold; text-decoration: none;">==</ins></div></td></tr>
</table>imported>RayAYanghttps://web.ma.utexas.edu/mediawiki/index.php?title=Free_boundary_problems&diff=368&oldid=previmported>RayAYang: Created page with "=Studied problems for $(-\Delta)^s$= Here are some free boundary problems for the fractional Laplace operator $(-\Delta)^s$ that have been studied to some extent. The [[extensi..."2011-05-24T02:42:35Z<p>Created page with "=Studied problems for $(-\Delta)^s$= Here are some free boundary problems for the fractional Laplace operator $(-\Delta)^s$ that have been studied to some extent. The [[extensi..."</p>
<p><b>New page</b></p><div>=Studied problems for $(-\Delta)^s$=<br />
Here are some free boundary problems for the fractional Laplace operator $(-\Delta)^s$ that have been studied to some extent. <br />
<br />
The [[extension technique]], representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below. <br />
<br />
*[[Fractional obstacle problem]]<br />
The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$. The problem can be formulated as a variational problem as well, either through the extension or through a [[Dirchlet form]] on $\mathbb{R}^n$. <br />
<br />
Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]]. <br />
<br />
*[[Fractional Alt-Caffarelli problem]]<br />
The problem is to seek the minimizer of an energy which is the sum of the Dirichlet form corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension. <br />
<br />
Like its 2nd order analogue, solutions to problem are known to have optimal regularity of Holder class $C^s$, and to be $s$-harmonic away from the zero set. They have nondegeneracy of order $s$ as well, which is to say that <br />
$$ u(X) \geq c d(X,\Lambda)^s $$<br />
where $\Lambda$ is the zero set of the $u$. <br />
<br />
Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat. <br />
<br />
*[[Fractional Alt-Phillips problem]]</div>imported>RayAYang