Nonlocal minimal surfaces and Monge Ampere equation: Difference between pages

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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 energy functional:
The Monge-Ampere equation refers to
\[ \det D^2 u = f(x). \]
The right hand side $f$ is always nonnegative. We look for a convex function $u$ which solves the equation.


\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]
The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator
\[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\]
is concave.


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 $E$.
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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.
 
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
 
\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]
 
 
== Nonlocal mean curvature ==
 
== Weak Nonlocal minimal surfaces ==
 
== Regularity results for Nonlocal minimal surfaces ==

Revision as of 16:46, 8 April 2015

The Monge-Ampere equation refers to \[ \det D^2 u = f(x). \] The right hand side $f$ is always nonnegative. We look for a convex function $u$ which solves the equation.

The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator \[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\] is concave.

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