Free boundary problems and Upcoming events: Difference between pages

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=Studied problems for $(-\Delta)^s$=
This is a list in chronological order of upcoming events that may be of interest to readers of this wiki. Feel free to add any events that you feel qualify!
Here are some free boundary problems for the fractional Laplace operator $(-\Delta)^s$ that have been studied to some extent.  


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.  
==2011==
*June 20-24 [http://math.utaustinportugal.org/summer011/index.phtml CoLab] workshop on Nonlinear PDEs at [http://www.math.ist.utl.pt/ Instituto Superior Técnico] in Lisbon, Portugal.
*June 27, 2011-July 1, 2011 [http://www.ma1.upc.edu/recerca/seminaris/JISD2011/indexjisd2011.html JISD 2011] at the [http://www.upc.edu/ Universitat Politècnica de Catalunya (UPC)] in Barcelona, Spain.


==[[Fractional obstacle problem]]==
==2012==
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 $\mathbb{R}^n$.  
*February 27-March 2, 2012: [http://www.ipam.ucla.edu/programs/pde2012/ Program in Nonlocal PDEs, Variational Problems and their Applications at IPAM]. Application deadline for funding support is January 2, 2012.
* May 2012: [http://math.uchicago.edu/~luis/special-month/index.html Special program in Chicago with minicourses in non local equations.] ([[Lecture notes of Silvestre's course]] available.)
* July 2012: [http://www.math.uni-bielefeld.de/nloc-school/ Summer School 2012 on Nonlocal Operators; Bielefeld University]
* July 2012: [http://www.math.uni-bielefeld.de/nloc2012/ Nonlocal Operators: Analysis, Probability, Geometry and Applications; Bielefeld University]


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]].
==2013==
 
*Milan, June 17-21: [http://www.mat.unimi.it/users/rocca/AdvPDE Recent advances in partial differential equations and applications]
==[[Fractional Alt-Caffarelli problem]]==
*Barcelona, July 15-19: [http://www.ma1.upc.edu/recerca/seminaris-recerca/jisd2013/syllabus13 11th Workshop on Interactions between Dynamical Systems and Partial Differential Equations]
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.
*Stanford, August 5-18: [http://math.stanford.edu/~ryzhik/SCHOOL-13/summer-school-13.html Recent Advances in PDEs and Fluids]
 
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
$$ u(X) \geq c d(X,\Lambda)^s $$
where $\Lambda$ is the zero set of the $u$.  
 
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.  
 
==[[Fractional Alt-Phillips problem]]==

Revision as of 14:07, 13 June 2013

This is a list in chronological order of upcoming events that may be of interest to readers of this wiki. Feel free to add any events that you feel qualify!

2011

2012

2013