(Difference between revisions)
 Revision as of 04:35, 7 February 2012 (view source)Nestor (Talk | contribs)← Older edit Latest revision as of 19:09, 23 September 2013 (view source)Tianling (Talk | contribs) (9 intermediate revisions not shown) Line 1: Line 1: __NOTOC__ __NOTOC__ + {{DISPLAYTITLE:{{FULLPAGENAME}}}} {| id="mp-topbanner" style="width:100%; background:#fcfcfc; margin-top:1.2em; border:1px solid #ccc;" {| id="mp-topbanner" style="width:100%; background:#fcfcfc; margin-top:1.2em; border:1px solid #ccc;" | style="width:56%; color:#000;" | | style="width:56%; color:#000;" | Line 11: Line 12: - 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]]. + 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 also keep a list of [[open problems]] and of [[upcoming events]]. 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. 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]]. |} |} Line 34: Line 35: * 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, the Paneitz operators $\mathcal{P}(s)$ encode information about the manifold, when the parameter is not an even integer they are fractional powers of the Laplacian, and therefore are nonlocal. + * 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 quasi-geostrophic equation]]. * In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasi-geostrophic 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 [[Hamilton-Jacobi 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 [[Hamilton-Jacobi equation with fractional diffusion]], [[conservation laws with fractional diffusion]], etc... Line 47: Line 49:
+ * [[Intro to nonlocal equations]] * [[Fractional Laplacian]] * [[Fractional Laplacian]] * [[Linear integro-differential operator]] * [[Linear integro-differential operator]] + + * [[Fully nonlinear integro-differential equations]] * [[Myths about nonlocal equations]] * [[Myths about nonlocal equations]] Line 59: Line 64: * [[Obstacle problem]] * [[Obstacle problem]] - |} |} |} |}

## Latest revision as of 19:09, 23 September 2013

 Welcome! This is the Nonlocal Equations Wiki (67 articles and counting)

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 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.

 Why nonlocal equations? 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 integro-differential 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 obstacle problem. 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 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, 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 quasi-geostrophic 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 Hamilton-Jacobi equation with fractional diffusion, conservation laws with fractional diffusion, etc...