Viscosity solutions and Category:Quasilinear equations: Difference between pages

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''Viscosity solutions'' are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations that is particularly suitable for the [[comparison principle]] to hold and for stability under uniform limits.
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


== Definition ==
\[ \mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
\[ u_t = \mbox{div} \left ( u^p \nabla u\right ) \]
\[ (-\Delta)^{s} u +H(x,u,\nabla u)=0 (2s>1) \]


The following is a definition of viscosity solutions for [[fully nonlinear integro-differential equations]] of the form $Iu=0$, assuming that the functions involved are defined in the whole space. Other situations can be considered with straight forward modifications in the definition. The nonlocal operator $I$ is not necessarily assumed to be translation invariant. All we assume about $I$ is that it is ''black box'' operator such that
In particular, note that all [[Semilinear equations]] are automatically quasilinear.
# $Iv(x_0)$ is well defined every time $v \in C^2(x_0)$.
# $Iv(x_0) \geq Iw(x_0)$ if $v(x_0)=w(x_0)$ and $v\geq w$ in $\R^n$.
 
An function $u : \R^n \to \R$, which is upper semicontinuous in $\Omega$, is said to be a '''viscosity subsolution''' in $\Omega$ if the following statement holds
<blockquote style="background:#DDDDDD;">
For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \geq u$ in $U$, we construct the auxiliary function $v$ as
\[ v(x) = \begin{cases}
\varphi(x) & \text{if } x \in U \\
u(x) & \text{if } x \notin U
\end{cases}\]
Then, we have the inequality \[ Iv(x_0) \geq 0. \]
</blockquote>
 
An function $u : \R^n \to \R$, which is lower semicontinuous in $\Omega$, is said to be a '''viscosity supersolution''' in $\Omega$ if the following statement holds
<blockquote style="background:#DDDDDD;">
For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \leq u$ in $U$, , we construct the auxiliary function $v$ as
\[ v(x) = \begin{cases}
\varphi(x) & \text{if } x \in U \\
u(x) & \text{if } x \notin U
\end{cases}\]
Then, we have the inequality \[ Iv(x_0) \leq 0. \]
</blockquote>
 
A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a '''viscosity solution'''.

Revision as of 17:12, 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

\[ \mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] \[ u_t = \mbox{div} \left ( u^p \nabla u\right ) \] \[ (-\Delta)^{s} u +H(x,u,\nabla u)=0 (2s>1) \]

In particular, note that all Semilinear equations are automatically quasilinear.