imported>Luis |
imported>Nestor |
| Line 1: |
Line 1: |
| 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 $L \in \mathcal L$.
| | 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) |
|
| |
|
| 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.
| | \[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] |
| | \[ u_t = \mbox{div} \left ( u^p \nabla u\right ) \] |
| | \[ u_t+(-\Delta)^{s} u +H(x,t,u,\nabla u)=0\;\; (2s>1) \] |
|
| |
|
| Given any family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
| | Equations which are not quasilinear are called [[Fully nonlinear]]. Note that all [[Semilinear equations]] are automatically quasilinear. |
| \begin{align*}
| |
| M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\
| |
| M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x)
| |
| \end{align*}
| |
| | |
| 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
| |
| \begin{align*}
| |
| P^+(D^2 u) &= \Lambda \ \mathrm{tr}(D^2u^+) - \lambda \ \mathrm{tr}(D^2u^-)\\
| |
| P^-(D^2 u) &= \lambda \ \mathrm{tr}(D^2u^+) - \Lambda \ \mathrm{tr}(D^2u^-)
| |
| \end{align*}
| |
| | |
| 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"/>
| |
| \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 \\
| |
| 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
| |
| \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".
| |
| | |
| == References ==
| |
| {{reflist|refs=
| |
| <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>
| |
| }}
| |
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)
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
\[ u_t = \mbox{div} \left ( u^p \nabla u\right ) \]
\[ u_t+(-\Delta)^{s} u +H(x,t,u,\nabla u)=0\;\; (2s>1) \]
Equations which are not quasilinear are called Fully nonlinear. Note that all Semilinear equations are automatically quasilinear.