Nonlocal porous medium equation and Viscosity solutions: 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. | |||
== 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 $\R^n$. | |||
and | 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 16:06, 2 June 2011
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
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 \[ Iv(x_0) \geq 0. \]
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
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 \[ Iv(x_0) \leq 0. \]
A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a viscosity solution.