From Mwiki
(→Nonlinear equations) 

(20 intermediate revisions not shown)  
Line 1:  Line 1:  
  +  __NOTOC__  
+  {{DISPLAYTITLE:<span style="display:none">{{FULLPAGENAME}}</span>}}  
+  { id="mptopbanner" style="width:100%; background:#fcfcfc; margintop:1.2em; border:1px solid #ccc;"  
+   style="width:56%; color:#000;"   
+  { style="width:280px; border:none; background:none;"  
+   style="width:280px; textalign:center; whitespace:nowrap; color:#000;"   
+  <div style="fontsize:162%; border:none; margin:0; padding:.1em; color:#000;">'''Welcome!'''</div>  
+  <div style="top:+0.2em; fontsize:95%;">This is the Nonlocal Equations Wiki</div>  
+  <div id="articlecount" style="width:100%; textalign:center; fontsize:85%;">([[Special:Statistics{{NUMBEROFARTICLES}}]] articles and counting)</div>  
+  }  
+    
  
  +  In this wiki we collect several results about nonlocal elliptic and parabolic equations. If you want to know what a nonlocal equation refers to, a good starting point would be the [[Intro to nonlocal equations]]. If you want to find information on a specific topic, you may want to check the [[list of equations]] or use the search option on the left.  
  We keep a list of [[open problems]] and  +  We also keep a list of [[open problems]] and of [[upcoming events]]. 
  +  The wiki has an assumed bias towards regularity results and consequently to equations for which some regularization occurs. But we also include some topics which are tangentially related, or even completely unrelated, to regularity.  
  All partial differential equations are a limit case of nonlocal equations  +  Some answers, including how to participate, can be found in the section about [[frequently asked questions]]. 
+  }  
+  
+  { id="mptopbanner" style="width:100%; background:#fcfcfc; margintop:1.2em; border:0px solid #ccc;"  
+   style="width:56%; color:#000;"   
+  
+  { id="mptopbanner" style="width:90%; background:#fcfcfc; margintop:1.2em; border:1px solid #ccc;"  
+   style="width:56%; color:#000;"   
+  
+  <div style="fontsize:150%; border:none; margin:0; padding:.1em; color:#000;">Why nonlocal equations?</div>  
+  
+  All partial differential equations are a limit case of nonlocal equations. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are  
* Optimal control problems with [[Levy processes]] give rise to the [[Bellman equation]], or in general any equation derived from jump processes will be some [[fully nonlinear integrodifferential equation]].  * Optimal control problems with [[Levy processes]] give rise to the [[Bellman equation]], or in general any equation derived from jump processes will be some [[fully nonlinear integrodifferential equation]].  
  * In [[financial mathematics]] it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the [[  +  * In [[financial mathematics]] it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the [[obstacle problem]]. 
* [[Nonlocal electrostatics]] is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.  * [[Nonlocal electrostatics]] is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.  
* The denoising algorithms in [[nonlocal image processing]] are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the [[nonlocal mean curvature flow]].  * The denoising algorithms in [[nonlocal image processing]] are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the [[nonlocal mean curvature flow]].  
* The [[Boltzmann equation]] models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified [[kinetic models]] can be used to derive the [[fractional heat equation]] without resorting to stochastic processes.  * The [[Boltzmann equation]] models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified [[kinetic models]] can be used to derive the [[fractional heat equation]] without resorting to stochastic processes.  
  * In conformal geometry,  +  * In conformal geometry, the [[conformally invariant operators]] encode information about the manifold. They include fractional powers of the Laplacian. 
* In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasigeostrophic equation]].  * In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasigeostrophic equation]].  
+  * Models for [[dislocation dynamics]] in crystals.  
* Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the [[Nonlocal porous medium equation]], the [[HamiltonJacobi equation with fractional diffusion]], [[conservation laws with fractional diffusion]], etc...  * Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the [[Nonlocal porous medium equation]], the [[HamiltonJacobi equation with fractional diffusion]], [[conservation laws with fractional diffusion]], etc...  
  ==  +  } 
  +  
+    
+  
+  { id="mptopbanner" style="width:40%; background:#fcfcfc; margintop:1.2em; border:1px solid #ccc;"  
+   style="width:56%; color:#000;"   
+  
+  <div style="fontsize:150%; border:none; margin:0; padding:.1em; color:#000;"> Suggested first reads </div>  
+  
+  * [[Intro to nonlocal equations]]  
+  
+  * [[Fractional Laplacian]]  
+  
+  * [[Linear integrodifferential operator]]  
  +  * [[Fully nonlinear integrodifferential equations]]  
  +  * [[Myths about nonlocal equations]]  
  +  * [[Surface quasigeostrophic equation]]  
  +  * [[Levy processes]]  
  +  
  +  * [[Obstacle problem]]  
  +  }  
  +  }  
  + 
Latest revision as of 19:09, 23 September 2013

We also keep a list of open problems and of upcoming events. The wiki has an assumed bias towards regularity results and consequently to equations for which some regularization occurs. But we also include some topics which are tangentially related, or even completely unrelated, to regularity. Some answers, including how to participate, can be found in the section about frequently asked questions. 

