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

Revision as of 17:27, 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). 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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-Broadly speaking, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that minimize (within a class of given admissible configurations) the following 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). For instance, the following equations are all quasilinear (and not semilinear)
-\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]
+\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
-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 $E$.
+<center> [[Mean curvature flow]] </center>
-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.
+\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]
-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> [[Nonlocal porous medium equation]] </center>
-\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]
+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.

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

Revision as of 17:27, 3 June 2011

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