Viscosity solutions - Revision history

imported>Luis at 15:21, 25 April 2012

2012-04-25T15:21:14Z

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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.		''Viscosity solutions'' are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations based on the notion of the [[comparison principle]]. It is especially suitable for stability under uniform limits.

	== Definition ==		== Definition ==

imported>Luis at 21:19, 2 June 2011

2011-06-02T21:19:16Z

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	A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a '''viscosity solution'''.		A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a '''viscosity solution'''.

			Note that if the operator $I$ happens to be local, the construction of the function $v$ is unnecessary since $Iv(x_0) = I\varphi(x_0)$. Thus for local equations the definition is given evaluating $I\varphi(x_0)$ instead.

imported>Luis: Created page with "''Viscosity solutions'' are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations that is particularly suitable for the [[comparison p..."

2011-06-02T21:06:19Z

Created page with "''Viscosity solutions'' are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations that is particularly suitable for the [[comparison p..."

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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.

== Definition ==

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
# $Iv(x_0)$ 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'''.