Nonlocal minimal surfaces and Category:Quasilinear equations: Difference between pages

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In broad and vague terms, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that are minimizers or critical points (within a class of given admissible configurations) of the energy functional:
A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not).  See also the article for [[Quasilinear equations]]


\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]


It can be checked easily that this agrees (save for a factor of $2$) with  norm of the characteristic function $\chi_E$ in the homogenous Sobolev space  $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$.
For instance, the following equations are all quasilinear (and not semilinear)


Classically,  [[minimal surfaces]] (or generally [[surfaces of constant mean curvature]] ) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart.
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]


Nonlocal minimal surfaces then, describe physical phenomena where the interaction potential does not decay fast enough as particles are apart, so that two particles on different phases and far from the interface still contribute a non-trivial amount to the total interaction energy, in particular, one may consider much more general energy functionals corresponding to different interaction potentials
<center> [[Mean curvature flow]] </center>


\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]  


== Definition ==
<center> [[Nonlocal porous medium equation]] </center>


Following the most accepted convention for [[minimal surfaces]],  a nonlocal minimal surface is the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\chi_E \in \dot{H}^{s/2}$ and whose [[Nonlocal mean curvature]] $H_s$ is identically zero (see next section), that is
Equations which are not quasilinear are called [[Fully nonlinear equations]], which include for instance [[Monge Ampére]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.


\[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\]
Note: In this category are listed all equations which are quasilinear  and NOT semilinear. Strictly speaking, all semilinear equations ought to be listed here aswell, however, as the specific methods and questions are so different in both categories (i.e. quasilinear techniques may give results for semilinear equations which are weaker  when compared to the more powerful methods tailor-made for semilinear ones), thus they are listed only in their own category.
 
In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$.
 
 
Example: Suppose that $E$ and $\Omega$ are such that for any other set $F$ such that $F \Delta E \subset \subset \Omega$ (i.e. $F$ agrees with $E$ outside $\Omega$) we have
 
\[J_s(E) \leq J_s(F) \]
 
Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$.
 
<div style="background:#DDEEFF;">
<blockquote>
'''Note''' For this definition to make sense, $\Sigma$ must be the boundary of some open set $E$, in this article, we will often refer to the set $E$ itself as "the" minimal surface, and no confusion should arise from this.  
</blockquote>
</div>
 
 
== Nonlocal mean curvature ==
 
== Surfaces minimizing non-local energy functionals ==
 
== The Caffarelli-Roquejoffre-Savin Regularity Theorem==

Revision as of 17:28, 3 June 2011

A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). See also the article for Quasilinear equations


For instance, the following equations are all quasilinear (and not semilinear)

\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]

Mean curvature flow

\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]

Nonlocal porous medium equation

Equations which are not quasilinear are called Fully nonlinear equations, which include for instance Monge Ampére and Fully nonlinear integro-differential equations. Note that all Semilinear equations are automatically quasilinear.

Note: In this category are listed all equations which are quasilinear and NOT semilinear. Strictly speaking, all semilinear equations ought to be listed here aswell, however, as the specific methods and questions are so different in both categories (i.e. quasilinear techniques may give results for semilinear equations which are weaker when compared to the more powerful methods tailor-made for semilinear ones), thus they are listed only in their own category.

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