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| The extremal operator associated to some class of linear operators $\mathcal{L}$ represent the maximal and minimal value that $Lu(x)$ can take from all possible choices of $L \in \mathcal L$.
| | 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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| The extremal operators are used to define [[uniformly elliptic|uniform ellipticity]] for nonlocal operators. In fact, the extremal operators are also the maximal and minimal nonlinear uniformly elliptic operators with respect to $\mathcal L$ that vanish at zero.
| | For instance, the following equations are all quasilinear (and the first two are NOT semilinear) |
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| Given any family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
| | \[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] |
| \begin{align*} | |
| M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
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| M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x)
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| \end{align*} | |
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| If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula
| | <center> [[Mean curvature flow]] </center> |
| \begin{align*}
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| P^+(D^2 u) &= \Lambda \ \mathrm{tr}(D^2u^+) - \lambda \ \mathrm{tr}(D^2u^-)\\
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| P^-(D^2 u) &= \lambda \ \mathrm{tr}(D^2u^+) - \Lambda \ \mathrm{tr}(D^2u^-)
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| \end{align*}
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| If $\mathcal{L}$ consists of all [[linear integro-differential operator|symmetric purely integro-differential operators, uniformly elliptic of order $s$]], then the extremal operators have the formula<ref name="S"/>
| | \[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \] |
| \begin{align*} | |
| M^+\, u &= \int_{\R^n} \left( \Lambda \delta u(x,y)^+ - \lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y \\
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| M^-\, u &= \int_{\R^n} \left( \lambda \delta u(x,y)^+ - \Lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y
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| \end{align*} | |
| where $\delta u(x,y) = (u(x+y) + u(x-y) - 2u(x))$. These two extremal operator are sometimes called "the ''monster'' Pucci operators".
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| == References ==
| | <center> [[Nonlocal porous medium equation]] </center> |
| {{reflist|refs=
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| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref> | |
| }}
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| [[Category:Fully nonlinear equations]] | | \[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\] |
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| | <center> Hamilton-Jacobi with fractional diffusion </center> |
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| | 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. |
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)
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
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
\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]
Hamilton-Jacobi with fractional diffusion
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.