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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:
| | Quasilinear equations are those which are linear in all terms except for the highest order derivatives (whether they are of fractional order or not). |
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| \[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]
| | For instance, the following equations are all quasilinear (and the first two are NOT semilinear) |
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| 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 $E$.
| | === Mean curvature flow === |
| | \[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] |
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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.
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| Nonlocal minimal surfaces then describe physical phenomena where the interaction potential does not decay fast enough as particles get farther and farther apart, so that two particles on different phases contribute a non-trivial amount to the total interaction energy even if they are away from the interface. In particular, one may consider much more general energy functionals corresponding to different interaction potentials | | === [[Nonlocal porous medium equation]] === |
| | \[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \] |
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| \[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]
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| == Definition == | | === [[Semilinear equations|Hamilton-Jacobi with fractional diffusion]] === |
| | \[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\] |
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| Following the most accepted convention for [[minimal surfaces]], a (classical) nonlocal minimal surface is (given $s\in (0,1)$) the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\Sigma$ is at least $C^{1,s+\epsilon}$ and more importantly,
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| \[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\]
| | Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called [[Fully nonlinear equations]], they include for instance the [[Monge Ampére Equation]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear. |
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| In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$. The quantity $H_s(x)$ is called the "Nonlocal mean curvature of order $s$ of $\Sigma$ at $x$", or briefly, "Nonlocal mean curvature".
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| 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
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| \[J_s(E) \leq J_s(F) \]
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| Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$.
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| <blockquote>
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| '''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.
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| </blockquote>
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| </div>
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| == Nonlocal mean curvature ==
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| == Surfaces minimizing non-local energy functionals ==
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| == The Caffarelli-Roquejoffre-Savin Regularity Theorem==
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Quasilinear equations are those which are linear in all terms except for the highest order derivatives (whether they are of fractional order or not).
For instance, the following equations are all quasilinear (and the first two are NOT semilinear)
Mean curvature flow
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]
\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]
Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called Fully nonlinear equations, they include for instance the Monge Ampére Equation and Fully nonlinear integro-differential equations. Note that all Semilinear equations are automatically quasilinear.
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